Properties

Label 6-9702e3-1.1-c1e3-0-4
Degree $6$
Conductor $913237656408$
Sign $-1$
Analytic cond. $464959.$
Root an. cond. $8.80175$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·4-s − 10·8-s + 3·11-s + 15·16-s − 3·17-s − 9·19-s − 9·22-s − 3·23-s − 9·29-s − 6·31-s − 21·32-s + 9·34-s + 21·37-s + 27·38-s − 12·41-s + 3·43-s + 18·44-s + 9·46-s − 3·47-s − 24·53-s + 27·58-s + 9·59-s − 6·61-s + 18·62-s + 28·64-s + 6·67-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s − 3.53·8-s + 0.904·11-s + 15/4·16-s − 0.727·17-s − 2.06·19-s − 1.91·22-s − 0.625·23-s − 1.67·29-s − 1.07·31-s − 3.71·32-s + 1.54·34-s + 3.45·37-s + 4.37·38-s − 1.87·41-s + 0.457·43-s + 2.71·44-s + 1.32·46-s − 0.437·47-s − 3.29·53-s + 3.54·58-s + 1.17·59-s − 0.768·61-s + 2.28·62-s + 7/2·64-s + 0.733·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 7^{6} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(464959.\)
Root analytic conductor: \(8.80175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9702} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 7^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + T )^{3} \)
3 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{3} \)
good5$D_{6}$ \( 1 + 4 p T^{3} + p^{3} T^{6} \)
13$C_2$ \( ( 1 + p T^{2} )^{3} \)
17$S_4\times C_2$ \( 1 + 3 T - 6 T^{2} - 37 T^{3} - 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 9 T + 69 T^{2} + 314 T^{3} + 69 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 3 T + 42 T^{2} + 49 T^{3} + 42 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 9 T + 39 T^{2} + 6 p T^{3} + 39 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 6 T + 45 T^{2} + 180 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 21 T + 243 T^{2} - 1782 T^{3} + 243 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 12 T + 96 T^{2} + 678 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 3 T - 3 T^{2} + 146 T^{3} - 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 3 T + 114 T^{2} + 273 T^{3} + 114 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{3} \)
59$S_4\times C_2$ \( 1 - 9 T + 129 T^{2} - 1014 T^{3} + 129 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 6 T + 120 T^{2} + 830 T^{3} + 120 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 6 T + 138 T^{2} - 592 T^{3} + 138 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 9 T + 105 T^{2} + 1170 T^{3} + 105 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 99 T^{2} - 480 T^{3} + 99 p T^{4} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 12 T + 270 T^{2} - 1880 T^{3} + 270 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 18 T + 282 T^{2} + 2824 T^{3} + 282 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 12 T + 255 T^{2} - 2120 T^{3} + 255 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{3} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.28788622642520136789915827878, −6.75473704140712739250575622704, −6.64802767496475947553437890589, −6.60529779673015006631654424414, −6.38002215852065879665967483075, −6.16323737919315567129194713919, −5.99440362039303322292340427587, −5.54224113410139393507225843392, −5.50352515419386436509365950406, −5.16328095151499299345461209047, −4.65303737481324065232243151987, −4.55350280690432991381603486417, −4.32701290164631969254751161653, −3.88192433243789002706517155410, −3.85590808501459984942544076550, −3.55821888774025826215802508992, −3.06322225371137151667421357932, −3.01080496136979650057679337074, −2.48662868946096378495441374598, −2.23857344486020674594146839751, −2.12249707919965707724894924317, −1.82692573456607302970413694887, −1.39126790606822179345833944557, −1.29764244572361468263513943308, −0.862623496353935634237203731785, 0, 0, 0, 0.862623496353935634237203731785, 1.29764244572361468263513943308, 1.39126790606822179345833944557, 1.82692573456607302970413694887, 2.12249707919965707724894924317, 2.23857344486020674594146839751, 2.48662868946096378495441374598, 3.01080496136979650057679337074, 3.06322225371137151667421357932, 3.55821888774025826215802508992, 3.85590808501459984942544076550, 3.88192433243789002706517155410, 4.32701290164631969254751161653, 4.55350280690432991381603486417, 4.65303737481324065232243151987, 5.16328095151499299345461209047, 5.50352515419386436509365950406, 5.54224113410139393507225843392, 5.99440362039303322292340427587, 6.16323737919315567129194713919, 6.38002215852065879665967483075, 6.60529779673015006631654424414, 6.64802767496475947553437890589, 6.75473704140712739250575622704, 7.28788622642520136789915827878

Graph of the $Z$-function along the critical line