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Properties

Label 4-832e2-1.1-c1e2-0-27
Degree $4$
Conductor $692224$
Sign $1$
Analytic cond. $44.1368$
Root an. cond. $2.57750$
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-s + 3·5-s − 3·7-s − 9-s + 4·11-s + 2·13-s + 3·15-s + 3·17-s + 12·19-s − 3·21-s + 25-s − 6·31-s + 4·33-s − 9·35-s + 7·37-s + 2·39-s + 6·41-s + 3·43-s − 3·45-s + 9·47-s − 3·49-s + 3·51-s + 18·53-s + 12·55-s + 12·57-s − 12·59-s − 2·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s − 1.13·7-s − 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.774·15-s + 0.727·17-s + 2.75·19-s − 0.654·21-s + 1/5·25-s − 1.07·31-s + 0.696·33-s − 1.52·35-s + 1.15·37-s + 0.320·39-s + 0.937·41-s + 0.457·43-s − 0.447·45-s + 1.31·47-s − 3/7·49-s + 0.420·51-s + 2.47·53-s + 1.61·55-s + 1.58·57-s − 1.56·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(692224\)    =    \(2^{12} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(44.1368\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 692224,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.524676212\)
\(L(\frac12)\) \(\approx\) \(3.524676212\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_c
5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_i
7$D_{4}$ \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_m
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
17$D_{4}$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_bg
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$D_{4}$ \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_cc
37$D_{4}$ \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_bw
41$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_cw
43$D_{4}$ \( 1 - 3 T - 18 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.43.ad_as
47$D_{4}$ \( 1 - 9 T + 76 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_cy
53$D_{4}$ \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.53.as_go
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$D_{4}$ \( 1 + 2 T - 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_abe
67$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.67.am_go
71$D_{4}$ \( 1 + 9 T + 124 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.71.j_eu
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.79.y_lq
83$D_{4}$ \( 1 + 10 T + 38 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.83.k_bm
89$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.89.a_eg
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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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

−10.11679133154778066153966609658, −9.806057451218276113604895515910, −9.501188052015337686833732691581, −9.381712764273865955572981583725, −8.942173863166352191410282609204, −8.450013455410768989159259017789, −7.85722718878998315629993807171, −7.31946970711597443624368235107, −7.06528923023124603922603022090, −6.48672463558548764138589872168, −5.88208074573168203198918741286, −5.66750605465859246248414450043, −5.49534100635074997241321141439, −4.54070247726807357277017577522, −3.71669901845409430887793003288, −3.62176695122971953067327633570, −2.77544523177250281345606381306, −2.59699132794889332705829526933, −1.47481550740806929509898867304, −1.02854606481196636854459325445, 1.02854606481196636854459325445, 1.47481550740806929509898867304, 2.59699132794889332705829526933, 2.77544523177250281345606381306, 3.62176695122971953067327633570, 3.71669901845409430887793003288, 4.54070247726807357277017577522, 5.49534100635074997241321141439, 5.66750605465859246248414450043, 5.88208074573168203198918741286, 6.48672463558548764138589872168, 7.06528923023124603922603022090, 7.31946970711597443624368235107, 7.85722718878998315629993807171, 8.450013455410768989159259017789, 8.942173863166352191410282609204, 9.381712764273865955572981583725, 9.501188052015337686833732691581, 9.806057451218276113604895515910, 10.11679133154778066153966609658

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