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Properties

Label 4-623808-1.1-c1e2-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 6·7-s − 8-s + 9-s − 12-s + 6·14-s + 16-s − 18-s + 4·19-s + 6·21-s + 24-s − 4·25-s − 27-s − 6·28-s + 8·29-s − 32-s + 36-s − 4·38-s − 14·41-s − 6·42-s + 8·43-s − 48-s + 14·49-s + 4·50-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 2.26·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.60·14-s + 1/4·16-s − 0.235·18-s + 0.917·19-s + 1.30·21-s + 0.204·24-s − 4/5·25-s − 0.192·27-s − 1.13·28-s + 1.48·29-s − 0.176·32-s + 1/6·36-s − 0.648·38-s − 2.18·41-s − 0.925·42-s + 1.21·43-s − 0.144·48-s + 2·49-s + 0.565·50-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: \(0\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4798670392\)
\(L(\frac12)\) \(\approx\) \(0.4798670392\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_1$ \( 1 + T \)
19$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
7$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.g_w
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.17.a_ba
23$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.23.a_au
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.ai_bm
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.31.a_ao
37$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.37.a_abe
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.o_ec
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.47.a_aq
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.c_du
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.59.as_hi
61$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.o_gg
67$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.67.a_aw
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ag_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.79.a_cs
83$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.83.a_as
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ae_eo
97$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.97.a_ak
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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

−8.343018439984546389257898993184, −8.031659682610873255204695144506, −7.28087395783489260352159508008, −7.06367193358233306020084017852, −6.56585551068497569527696059127, −6.21301993726660510715823951255, −5.91316304692042311134144103720, −5.23167748159334541864511223091, −4.77961749935921841231192928719, −3.84400345747102080111828166468, −3.54084738699049371014332404871, −2.93699818054910732426911985544, −2.41901334891264587250948678341, −1.35468836201213068672382672790, −0.42501605703668058217811839912, 0.42501605703668058217811839912, 1.35468836201213068672382672790, 2.41901334891264587250948678341, 2.93699818054910732426911985544, 3.54084738699049371014332404871, 3.84400345747102080111828166468, 4.77961749935921841231192928719, 5.23167748159334541864511223091, 5.91316304692042311134144103720, 6.21301993726660510715823951255, 6.56585551068497569527696059127, 7.06367193358233306020084017852, 7.28087395783489260352159508008, 8.031659682610873255204695144506, 8.343018439984546389257898993184

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