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Properties

Label 4-325e2-1.1-c1e2-0-11
Degree $4$
Conductor $105625$
Sign $1$
Analytic cond. $6.73474$
Root an. cond. $1.61094$
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  + 4·4-s + 5·9-s − 12·11-s + 12·16-s + 8·19-s + 6·29-s − 8·31-s + 20·36-s + 12·41-s − 48·44-s − 2·49-s + 12·59-s − 2·61-s + 32·64-s − 12·71-s + 32·76-s − 22·79-s + 16·81-s − 60·99-s − 30·101-s − 4·109-s + 24·116-s + 86·121-s − 32·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 2·4-s + 5/3·9-s − 3.61·11-s + 3·16-s + 1.83·19-s + 1.11·29-s − 1.43·31-s + 10/3·36-s + 1.87·41-s − 7.23·44-s − 2/7·49-s + 1.56·59-s − 0.256·61-s + 4·64-s − 1.42·71-s + 3.67·76-s − 2.47·79-s + 16/9·81-s − 6.03·99-s − 2.98·101-s − 0.383·109-s + 2.22·116-s + 7.81·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(105625\)    =    \(5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(6.73474\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 105625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.522027340\)
\(L(\frac12)\) \(\approx\) \(2.522027340\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.2.a_ae
3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.23.a_abl
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.43.a_abl
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.53.a_az
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.61.c_et
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.67.a_ck
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.79.w_kt
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−11.72804645703094848947143972813, −11.28056960963013248711479131282, −10.87570316973455003894197143237, −10.38532992459382960078819564760, −10.17886255147936818805253913674, −9.942892024981197959889933819692, −9.155262343305100798286275974271, −8.083720860975592525034512941519, −7.85186770058065637042496647655, −7.57271774595739394408636961578, −7.07307005488618920370399863771, −6.84822816718476129432659610580, −5.70478599014574589119459369342, −5.61468073661378941204387736572, −5.07024791076197818923230671886, −4.21315269890944364716835223631, −3.15058073148390548229671054324, −2.78273445202813028289465280478, −2.22500974514134351030333953752, −1.26623473151111815808494300679, 1.26623473151111815808494300679, 2.22500974514134351030333953752, 2.78273445202813028289465280478, 3.15058073148390548229671054324, 4.21315269890944364716835223631, 5.07024791076197818923230671886, 5.61468073661378941204387736572, 5.70478599014574589119459369342, 6.84822816718476129432659610580, 7.07307005488618920370399863771, 7.57271774595739394408636961578, 7.85186770058065637042496647655, 8.083720860975592525034512941519, 9.155262343305100798286275974271, 9.942892024981197959889933819692, 10.17886255147936818805253913674, 10.38532992459382960078819564760, 10.87570316973455003894197143237, 11.28056960963013248711479131282, 11.72804645703094848947143972813

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