Properties

Label 4-1170e2-1.1-c1e2-0-0
Degree $4$
Conductor $1368900$
Sign $1$
Analytic cond. $87.2822$
Root an. cond. $3.05654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·5-s + 2·7-s + 8-s + 2·10-s − 5·11-s − 2·13-s − 2·14-s − 16-s − 2·17-s + 2·19-s + 5·22-s − 23-s + 3·25-s + 2·26-s + 5·29-s − 22·31-s + 2·34-s − 4·35-s − 3·37-s − 2·38-s − 2·40-s − 2·41-s + 11·43-s + 46-s − 18·47-s + 7·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.894·5-s + 0.755·7-s + 0.353·8-s + 0.632·10-s − 1.50·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s − 0.485·17-s + 0.458·19-s + 1.06·22-s − 0.208·23-s + 3/5·25-s + 0.392·26-s + 0.928·29-s − 3.95·31-s + 0.342·34-s − 0.676·35-s − 0.493·37-s − 0.324·38-s − 0.316·40-s − 0.312·41-s + 1.67·43-s + 0.147·46-s − 2.62·47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1368900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(87.2822\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1170} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1368900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06038143620\)
\(L(\frac12)\) \(\approx\) \(0.06038143620\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.00951160785558540183002795556, −9.473859665148937651333239825775, −9.098965077468531368962901718987, −8.723127140811028791652340030552, −8.415407784832923272019597645413, −7.75253862867312463914239984826, −7.67098409007848145485831192570, −7.24109723518160922069449465702, −7.13387812561495646987171325234, −6.06101002514470061891662554653, −5.86229929031065256797730077483, −5.17177699702265889004474104333, −4.76826011019887716641288391011, −4.54621397349729113012302837870, −3.87873745372381907813598431733, −3.06420844187965613381915352231, −2.96005807308468652889340804570, −1.82967052604062839174545404150, −1.60420868120687565640930258082, −0.11725664343122062512449796391, 0.11725664343122062512449796391, 1.60420868120687565640930258082, 1.82967052604062839174545404150, 2.96005807308468652889340804570, 3.06420844187965613381915352231, 3.87873745372381907813598431733, 4.54621397349729113012302837870, 4.76826011019887716641288391011, 5.17177699702265889004474104333, 5.86229929031065256797730077483, 6.06101002514470061891662554653, 7.13387812561495646987171325234, 7.24109723518160922069449465702, 7.67098409007848145485831192570, 7.75253862867312463914239984826, 8.415407784832923272019597645413, 8.723127140811028791652340030552, 9.098965077468531368962901718987, 9.473859665148937651333239825775, 10.00951160785558540183002795556

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