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Properties

Label 4-47916-1.1-c1e2-0-0
Degree $4$
Conductor $47916$
Sign $1$
Analytic cond. $3.05516$
Root an. cond. $1.32208$
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  − 2·3-s + 4-s + 4·5-s + 3·9-s − 11-s − 2·12-s − 8·15-s + 16-s + 4·20-s + 8·23-s + 2·25-s − 4·27-s + 2·33-s + 3·36-s + 12·37-s − 44-s + 12·45-s − 24·47-s − 2·48-s + 2·49-s + 4·53-s − 4·55-s + 24·59-s − 8·60-s + 64-s + 8·67-s − 16·69-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 1.78·5-s + 9-s − 0.301·11-s − 0.577·12-s − 2.06·15-s + 1/4·16-s + 0.894·20-s + 1.66·23-s + 2/5·25-s − 0.769·27-s + 0.348·33-s + 1/2·36-s + 1.97·37-s − 0.150·44-s + 1.78·45-s − 3.50·47-s − 0.288·48-s + 2/7·49-s + 0.549·53-s − 0.539·55-s + 3.12·59-s − 1.03·60-s + 1/8·64-s + 0.977·67-s − 1.92·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(47916\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(3.05516\)
Root analytic conductor: \(1.32208\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 47916,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.458275431\)
\(L(\frac12)\) \(\approx\) \(1.458275431\)
\(L(\frac{3}{2})\) 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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.a_acw
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−10.17335307448534281740885152086, −9.626249861423627809345281403043, −9.540951208346174452410797239179, −8.638085568255906260665351447274, −8.036115207635588683933447181084, −7.24765397306950965920878295914, −6.83452626373758769313406188857, −6.15685276163708486295790955262, −6.00719019897905662750930944335, −5.20400804389664174185532774012, −5.03220048416044134767486652990, −4.00861335496683547477816414246, −2.90847219285821389032814298970, −2.14568161285268378115047856310, −1.25132340594816139878900815015, 1.25132340594816139878900815015, 2.14568161285268378115047856310, 2.90847219285821389032814298970, 4.00861335496683547477816414246, 5.03220048416044134767486652990, 5.20400804389664174185532774012, 6.00719019897905662750930944335, 6.15685276163708486295790955262, 6.83452626373758769313406188857, 7.24765397306950965920878295914, 8.036115207635588683933447181084, 8.638085568255906260665351447274, 9.540951208346174452410797239179, 9.626249861423627809345281403043, 10.17335307448534281740885152086

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