Properties

Label 4-95e2-1.1-c1e2-0-1
Degree $4$
Conductor $9025$
Sign $1$
Analytic cond. $0.575441$
Root an. cond. $0.870964$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 3·5-s + 9-s + 6·11-s − 19-s − 6·20-s + 4·25-s − 3·29-s − 2·31-s + 2·36-s − 6·41-s + 12·44-s − 3·45-s + 2·49-s − 18·55-s − 15·59-s + 19·61-s − 8·64-s − 9·71-s − 2·76-s + 79-s − 8·81-s − 27·89-s + 3·95-s + 6·99-s + 8·100-s − 3·101-s + ⋯
L(s)  = 1  + 4-s − 1.34·5-s + 1/3·9-s + 1.80·11-s − 0.229·19-s − 1.34·20-s + 4/5·25-s − 0.557·29-s − 0.359·31-s + 1/3·36-s − 0.937·41-s + 1.80·44-s − 0.447·45-s + 2/7·49-s − 2.42·55-s − 1.95·59-s + 2.43·61-s − 64-s − 1.06·71-s − 0.229·76-s + 0.112·79-s − 8/9·81-s − 2.86·89-s + 0.307·95-s + 0.603·99-s + 4/5·100-s − 0.298·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.575441\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.037771598\)
\(L(\frac12)\) \(\approx\) \(1.037771598\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 3 T + p T^{2} \)
19$C_2$ \( 1 + T + p T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.a_ax
17$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.17.a_b
23$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.23.a_ar
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.d_bo
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.c_bb
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.43.a_h
47$C_2^2$ \( 1 + 79 T^{2} + p^{2} T^{4} \) 2.47.a_db
53$C_2^2$ \( 1 + 73 T^{2} + p^{2} T^{4} \) 2.53.a_cv
59$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.p_fy
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.61.at_hk
67$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.67.a_bf
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.j_fm
73$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.73.a_abj
79$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.ab_fi
83$C_2^2$ \( 1 + 142 T^{2} + p^{2} T^{4} \) 2.83.a_fm
89$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.bb_nc
97$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.97.a_n
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.58692691133839009958423248535, −11.27702269848948999971040850951, −10.75280211293488605551414163339, −9.973598780879133927112653670344, −9.312001046862536482265341500143, −8.705496902004465310127456483391, −8.119737263540234289574514989364, −7.31927275417342389051953674119, −6.97104040790960408210140243816, −6.47446141033857712132803146990, −5.61761072837646834772052306310, −4.41273927766682549439809116528, −3.96569208535790183313071922941, −3.10986503405460942944420675135, −1.69653871394278332647107205445, 1.69653871394278332647107205445, 3.10986503405460942944420675135, 3.96569208535790183313071922941, 4.41273927766682549439809116528, 5.61761072837646834772052306310, 6.47446141033857712132803146990, 6.97104040790960408210140243816, 7.31927275417342389051953674119, 8.119737263540234289574514989364, 8.705496902004465310127456483391, 9.312001046862536482265341500143, 9.973598780879133927112653670344, 10.75280211293488605551414163339, 11.27702269848948999971040850951, 11.58692691133839009958423248535

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