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Properties

Label 4-837e2-1.1-c1e2-0-9
Degree $4$
Conductor $700569$
Sign $1$
Analytic cond. $44.6688$
Root an. cond. $2.58524$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 4·5-s + 5·7-s + 3·8-s − 4·10-s + 4·11-s + 7·13-s + 5·14-s + 16-s − 3·17-s + 3·19-s − 4·20-s + 4·22-s − 8·23-s + 2·25-s + 7·26-s + 5·28-s − 2·29-s + 2·31-s − 32-s − 3·34-s − 20·35-s − 37-s + 3·38-s − 12·40-s + 20·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.78·5-s + 1.88·7-s + 1.06·8-s − 1.26·10-s + 1.20·11-s + 1.94·13-s + 1.33·14-s + 1/4·16-s − 0.727·17-s + 0.688·19-s − 0.894·20-s + 0.852·22-s − 1.66·23-s + 2/5·25-s + 1.37·26-s + 0.944·28-s − 0.371·29-s + 0.359·31-s − 0.176·32-s − 0.514·34-s − 3.38·35-s − 0.164·37-s + 0.486·38-s − 1.89·40-s + 3.12·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(700569\)    =    \(3^{6} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(44.6688\)
Root analytic conductor: \(2.58524\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 700569,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.618906411\)
\(L(\frac12)\) \(\approx\) \(3.618906411\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad3 \( 1 \)
31$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \) 2.2.ab_a
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$D_{4}$ \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.7.af_q
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$D_{4}$ \( 1 - 7 T + 34 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.13.ah_bi
17$D_{4}$ \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_bg
19$D_{4}$ \( 1 - 3 T + 36 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.19.ad_bk
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_bq
37$D_{4}$ \( 1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4} \) 2.37.b_bk
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$D_{4}$ \( 1 - 7 T + 94 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.43.ah_dq
47$D_{4}$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) 2.47.b_am
53$D_{4}$ \( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_ea
59$D_{4}$ \( 1 + 9 T + 134 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.59.j_fe
61$D_{4}$ \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.61.d_s
67$D_{4}$ \( 1 + 11 T + 126 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.67.l_ew
71$D_{4}$ \( 1 + 17 T + 210 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.71.r_ic
73$D_{4}$ \( 1 - 5 T - 56 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.73.af_ace
79$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.79.ao_hz
83$D_{4}$ \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_gc
89$D_{4}$ \( 1 - 7 T + 152 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.89.ah_fw
97$D_{4}$ \( 1 + 8 T + 193 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_hl
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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

−10.79224460129734108447869831474, −10.23036371845801312239391634134, −9.404700205727735841249837963326, −9.051585784897843585282778962400, −8.559062221028722746882991568688, −8.168458345567940119074118004922, −7.80541640538875344800167567787, −7.42926379379352767430567103378, −7.31902090549693939201455723213, −6.38954549478893384728467389676, −5.92663481450136159842645464249, −5.75974207936197301028294972534, −4.64905313043738119371212516944, −4.45446774594436266310557508607, −4.19963635739187223039725560909, −3.79856750216524231377379740804, −3.29414563104434147160555499651, −2.13830428910729851363638158476, −1.65562079168628561815037349514, −0.930901099224452430205058629520, 0.930901099224452430205058629520, 1.65562079168628561815037349514, 2.13830428910729851363638158476, 3.29414563104434147160555499651, 3.79856750216524231377379740804, 4.19963635739187223039725560909, 4.45446774594436266310557508607, 4.64905313043738119371212516944, 5.75974207936197301028294972534, 5.92663481450136159842645464249, 6.38954549478893384728467389676, 7.31902090549693939201455723213, 7.42926379379352767430567103378, 7.80541640538875344800167567787, 8.168458345567940119074118004922, 8.559062221028722746882991568688, 9.051585784897843585282778962400, 9.404700205727735841249837963326, 10.23036371845801312239391634134, 10.79224460129734108447869831474

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