Properties

Label 4-312e2-1.1-c1e2-0-35
Degree $4$
Conductor $97344$
Sign $1$
Analytic cond. $6.20673$
Root an. cond. $1.57839$
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  + 3-s + 6·5-s + 4·7-s − 4·11-s − 5·13-s + 6·15-s − 3·17-s + 4·19-s + 4·21-s + 8·23-s + 17·25-s − 27-s + 5·29-s − 16·31-s − 4·33-s + 24·35-s − 7·37-s − 5·39-s + 9·41-s − 8·43-s − 8·47-s + 7·49-s − 3·51-s − 10·53-s − 24·55-s + 4·57-s − 4·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 2.68·5-s + 1.51·7-s − 1.20·11-s − 1.38·13-s + 1.54·15-s − 0.727·17-s + 0.917·19-s + 0.872·21-s + 1.66·23-s + 17/5·25-s − 0.192·27-s + 0.928·29-s − 2.87·31-s − 0.696·33-s + 4.05·35-s − 1.15·37-s − 0.800·39-s + 1.40·41-s − 1.21·43-s − 1.16·47-s + 49-s − 0.420·51-s − 1.37·53-s − 3.23·55-s + 0.529·57-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(97344\)    =    \(2^{6} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(6.20673\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 97344,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.072958397\)
\(L(\frac12)\) \(\approx\) \(3.072958397\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ae_j
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_f
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_ad
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bp
29$C_2^2$ \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.29.af_ae
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.37.h_m
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_bo
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.47.i_eg
53$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.53.k_fb
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_abr
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.61.af_abk
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_ad
71$C_2^2$ \( 1 - 4 T - 55 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_acd
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.ao_dv
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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

−11.97203891351371068805972167806, −11.13206431393542394912977985172, −10.88746238257520542034844996986, −10.58718193668514673289818130812, −9.746521821197915874565769412383, −9.721404877693394660012466115518, −9.109819490646681994508145771921, −8.893177391750017321594638484443, −7.915936716857834631976966885766, −7.87525592776815281158952278370, −6.87345100907401027477054771786, −6.80747266974240587771543573211, −5.56754534870062268907340746091, −5.53061107605458742488750484563, −5.00051044922080561458393268027, −4.68413657355429970066777703793, −3.24299268288688182379717532262, −2.61987270067235914312200550890, −1.97972979252517803365857623707, −1.65252734159892672478220869023, 1.65252734159892672478220869023, 1.97972979252517803365857623707, 2.61987270067235914312200550890, 3.24299268288688182379717532262, 4.68413657355429970066777703793, 5.00051044922080561458393268027, 5.53061107605458742488750484563, 5.56754534870062268907340746091, 6.80747266974240587771543573211, 6.87345100907401027477054771786, 7.87525592776815281158952278370, 7.915936716857834631976966885766, 8.893177391750017321594638484443, 9.109819490646681994508145771921, 9.721404877693394660012466115518, 9.746521821197915874565769412383, 10.58718193668514673289818130812, 10.88746238257520542034844996986, 11.13206431393542394912977985172, 11.97203891351371068805972167806

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