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Properties

Label 4-623808-1.1-c1e2-0-35
Degree $4$
Conductor $623808$
Sign $-1$
Analytic cond. $39.7745$
Root an. cond. $2.51131$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 6-s − 4·7-s − 3·8-s + 9-s + 12-s − 4·14-s − 16-s + 18-s − 2·19-s + 4·21-s + 3·24-s + 6·25-s − 27-s + 4·28-s − 10·29-s + 5·32-s − 36-s − 2·38-s + 2·41-s + 4·42-s + 12·43-s + 48-s − 2·49-s + 6·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 1.06·14-s − 1/4·16-s + 0.235·18-s − 0.458·19-s + 0.872·21-s + 0.612·24-s + 6/5·25-s − 0.192·27-s + 0.755·28-s − 1.85·29-s + 0.883·32-s − 1/6·36-s − 0.324·38-s + 0.312·41-s + 0.617·42-s + 1.82·43-s + 0.144·48-s − 2/7·49-s + 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(623808\)    =    \(2^{6} \cdot 3^{3} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(39.7745\)
Root analytic conductor: \(2.51131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( 1 + T \)
19$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.k_cg
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.37.a_abm
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ac_de
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.43.am_eo
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.47.a_as
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.53.ac_ec
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ae_eo
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.61.aq_ha
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ai_fm
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.79.a_by
83$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.83.a_by
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ao_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

−8.026788032680766408494931267817, −7.75964323250008018478458553583, −6.95506461508138895874463255855, −6.68959812679724865382932359605, −6.35080752020181678773537210788, −5.71077290534709305134481876861, −5.48589863444284288733534494699, −4.96864569759886266436590334575, −4.31099994783749933785688411911, −3.83601682084430760741688284180, −3.50691369660934219119163853261, −2.81253754681733287219030073863, −2.22664346453673729573198760718, −0.925750835916661847382142578094, 0, 0.925750835916661847382142578094, 2.22664346453673729573198760718, 2.81253754681733287219030073863, 3.50691369660934219119163853261, 3.83601682084430760741688284180, 4.31099994783749933785688411911, 4.96864569759886266436590334575, 5.48589863444284288733534494699, 5.71077290534709305134481876861, 6.35080752020181678773537210788, 6.68959812679724865382932359605, 6.95506461508138895874463255855, 7.75964323250008018478458553583, 8.026788032680766408494931267817

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