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Properties

Label 4-25975-1.1-c1e2-0-1
Degree $4$
Conductor $25975$
Sign $1$
Analytic cond. $1.65618$
Root an. cond. $1.13442$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 5-s + 4·9-s + 5·11-s − 3·16-s − 4·19-s − 20-s − 4·25-s + 10·29-s − 3·31-s − 4·36-s − 5·41-s − 5·44-s + 4·45-s − 2·49-s + 5·55-s + 17·59-s + 3·61-s + 7·64-s − 3·71-s + 4·76-s − 3·79-s − 3·80-s + 7·81-s − 3·89-s − 4·95-s + 20·99-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.447·5-s + 4/3·9-s + 1.50·11-s − 3/4·16-s − 0.917·19-s − 0.223·20-s − 4/5·25-s + 1.85·29-s − 0.538·31-s − 2/3·36-s − 0.780·41-s − 0.753·44-s + 0.596·45-s − 2/7·49-s + 0.674·55-s + 2.21·59-s + 0.384·61-s + 7/8·64-s − 0.356·71-s + 0.458·76-s − 0.337·79-s − 0.335·80-s + 7/9·81-s − 0.317·89-s − 0.410·95-s + 2.01·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25975 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25975 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25975\)    =    \(5^{2} \cdot 1039\)
Sign: $1$
Analytic conductor: \(1.65618\)
Root analytic conductor: \(1.13442\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25975,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.320016936\)
\(L(\frac12)\) \(\approx\) \(1.320016936\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad5$C_2$ \( 1 - T + p T^{2} \)
1039$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 50 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.af_w
13$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \) 2.13.a_j
17$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.17.a_h
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.e_bm
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.29.ak_cw
31$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.d_cm
37$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.37.a_ao
41$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.f_bg
43$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.43.a_acj
47$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \) 2.47.a_abx
53$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.53.a_ca
59$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.59.ar_hc
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.61.ad_de
67$C_2^2$ \( 1 + 119 T^{2} + p^{2} T^{4} \) 2.67.a_ep
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.71.d_fm
73$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.73.a_cc
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.79.d_cs
83$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.83.a_n
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.d_gw
97$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.97.a_ay
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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

−10.46722450863992337020917536076, −10.05000567554588174898371424537, −9.574775262783048092692509925848, −9.171420253392451387349210653532, −8.532243583080618124841738830053, −8.148717557150923349142579639558, −7.11657760120690900499287669017, −6.77920880729772679392762182116, −6.37688224410657248976187246095, −5.51901439642800394456939325845, −4.68151735857002214554870619023, −4.20912317314450573412989060946, −3.68998071003431901536202757144, −2.33378565240791836213367563097, −1.37244404615155255266156001144, 1.37244404615155255266156001144, 2.33378565240791836213367563097, 3.68998071003431901536202757144, 4.20912317314450573412989060946, 4.68151735857002214554870619023, 5.51901439642800394456939325845, 6.37688224410657248976187246095, 6.77920880729772679392762182116, 7.11657760120690900499287669017, 8.148717557150923349142579639558, 8.532243583080618124841738830053, 9.171420253392451387349210653532, 9.574775262783048092692509925848, 10.05000567554588174898371424537, 10.46722450863992337020917536076

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