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Properties

Label 4-5100e2-1.1-c1e2-0-16
Degree $4$
Conductor $26010000$
Sign $1$
Analytic cond. $1658.42$
Root an. cond. $6.38151$
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  − 9-s + 6·11-s − 2·19-s − 10·29-s + 12·31-s + 10·41-s + 13·49-s − 8·59-s − 4·61-s + 24·71-s − 12·79-s + 81-s + 8·89-s − 6·99-s + 28·101-s + 12·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  − 1/3·9-s + 1.80·11-s − 0.458·19-s − 1.85·29-s + 2.15·31-s + 1.56·41-s + 13/7·49-s − 1.04·59-s − 0.512·61-s + 2.84·71-s − 1.35·79-s + 1/9·81-s + 0.847·89-s − 0.603·99-s + 2.78·101-s + 1.14·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26010000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1658.42\)
Root analytic conductor: \(6.38151\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 26010000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.572768794\)
\(L(\frac12)\) \(\approx\) \(3.572768794\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
17$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.7.a_an
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.23.a_abq
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.29.k_df
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.31.am_du
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.37.a_acv
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.41.ak_ed
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 + 75 T^{2} + p^{2} T^{4} \) 2.47.a_cx
53$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.53.a_p
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.73.a_x
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.79.m_hm
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.89.ai_hm
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

−8.467132703527706988056431842422, −8.008344353199177480174596600086, −7.60841969027613344442237919350, −7.46157970889129836355595890886, −6.89626654278523870356782778648, −6.54139208937578183844298365969, −6.19970249914630084247988314563, −6.10257358970966524742627372640, −5.42648218827654808814157414637, −5.30151636227306976678832792527, −4.46465427005546982190048508354, −4.37017207054826590867799794357, −4.01922700746083951469093353481, −3.52179719520933048222143004110, −3.16181846924904714075739091233, −2.59274835567855475876969783960, −2.08800904308444097327123338874, −1.71817045535855694975864207622, −0.945959197102127054280674007046, −0.62263138106937022346449939365, 0.62263138106937022346449939365, 0.945959197102127054280674007046, 1.71817045535855694975864207622, 2.08800904308444097327123338874, 2.59274835567855475876969783960, 3.16181846924904714075739091233, 3.52179719520933048222143004110, 4.01922700746083951469093353481, 4.37017207054826590867799794357, 4.46465427005546982190048508354, 5.30151636227306976678832792527, 5.42648218827654808814157414637, 6.10257358970966524742627372640, 6.19970249914630084247988314563, 6.54139208937578183844298365969, 6.89626654278523870356782778648, 7.46157970889129836355595890886, 7.60841969027613344442237919350, 8.008344353199177480174596600086, 8.467132703527706988056431842422

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