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Properties

Label 4-637e2-1.1-c1e2-0-2
Degree $4$
Conductor $405769$
Sign $1$
Analytic cond. $25.8721$
Root an. cond. $2.25532$
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  + 3·2-s − 4·3-s + 4·4-s − 3·5-s − 12·6-s + 3·8-s + 6·9-s − 9·10-s − 16·12-s + 5·13-s + 12·15-s + 3·16-s − 3·17-s + 18·18-s − 12·20-s + 6·23-s − 12·24-s + 25-s + 15·26-s + 4·27-s − 3·29-s + 36·30-s − 6·31-s + 6·32-s − 9·34-s + 24·36-s − 15·37-s + ⋯
L(s)  = 1  + 2.12·2-s − 2.30·3-s + 2·4-s − 1.34·5-s − 4.89·6-s + 1.06·8-s + 2·9-s − 2.84·10-s − 4.61·12-s + 1.38·13-s + 3.09·15-s + 3/4·16-s − 0.727·17-s + 4.24·18-s − 2.68·20-s + 1.25·23-s − 2.44·24-s + 1/5·25-s + 2.94·26-s + 0.769·27-s − 0.557·29-s + 6.57·30-s − 1.07·31-s + 1.06·32-s − 1.54·34-s + 4·36-s − 2.46·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(405769\)    =    \(7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(25.8721\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 405769,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.129235079\)
\(L(\frac12)\) \(\approx\) \(1.129235079\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad7 \( 1 \)
13$C_2$ \( 1 - 5 T + p T^{2} \)
good2$C_2^2$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.2.ad_f
3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.3.e_k
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_i
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_au
31$C_2^2$ \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_br
37$C_2^2$ \( 1 + 15 T + 112 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.37.p_ei
41$C_2^2$ \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_cq
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2^2$ \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_ch
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_abs
59$C_2^2$ \( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_ed
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.61.c_et
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.a_aes
71$C_2^2$ \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_df
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.d_cy
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.79.e_acl
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.83.a_ba
89$C_2^2$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_fh
97$C_2^2$ \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_fp
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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

−11.08060364416739305060101066546, −10.93638211188093199515334930785, −10.40199763930213594460594816242, −9.733508496049652972552692847022, −8.925839659442933591879481987419, −8.505161975160933598036068709853, −8.153428390491766762025414323547, −7.26709058866778054975839607403, −6.85122568022402587135816121391, −6.68201826740585785137785132156, −5.84064458120069773276357046879, −5.77344741386487776485095227481, −5.13663646496709187077605882939, −5.08996316171099503409157474381, −4.18916674648145817787085134914, −4.16319865780015139198901909489, −3.41830582243927935744408857356, −3.05364108688510757743403718478, −1.62686852176340153291936467398, −0.48842600497963476640525270221, 0.48842600497963476640525270221, 1.62686852176340153291936467398, 3.05364108688510757743403718478, 3.41830582243927935744408857356, 4.16319865780015139198901909489, 4.18916674648145817787085134914, 5.08996316171099503409157474381, 5.13663646496709187077605882939, 5.77344741386487776485095227481, 5.84064458120069773276357046879, 6.68201826740585785137785132156, 6.85122568022402587135816121391, 7.26709058866778054975839607403, 8.153428390491766762025414323547, 8.505161975160933598036068709853, 8.925839659442933591879481987419, 9.733508496049652972552692847022, 10.40199763930213594460594816242, 10.93638211188093199515334930785, 11.08060364416739305060101066546

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