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Properties

Label 4-623808-1.1-c1e2-0-47
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  − 2·2-s + 3-s + 2·4-s − 2·6-s + 9-s − 2·11-s + 2·12-s − 4·16-s + 2·17-s − 2·18-s + 4·22-s − 4·25-s + 27-s + 8·32-s − 2·33-s − 4·34-s + 2·36-s − 8·43-s − 4·44-s − 4·48-s + 2·49-s + 8·50-s + 2·51-s − 2·54-s − 8·59-s − 8·64-s + 4·66-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 16-s + 0.485·17-s − 0.471·18-s + 0.852·22-s − 4/5·25-s + 0.192·27-s + 1.41·32-s − 0.348·33-s − 0.685·34-s + 1/3·36-s − 1.21·43-s − 0.603·44-s − 0.577·48-s + 2/7·49-s + 1.13·50-s + 0.280·51-s − 0.272·54-s − 1.04·59-s − 64-s + 0.492·66-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$C_2$ \( 1 + p T + p T^{2} \)
3$C_1$ \( 1 - T \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.c_o
13$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.13.a_q
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.ac_k
23$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.23.a_ae
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.31.a_au
37$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.37.a_bo
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.i_bm
47$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.47.a_au
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.i_eo
61$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.61.a_ada
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.71.a_aw
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.73.ay_ko
79$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.79.a_ae
83$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.k_hi
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.89.am_hq
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.ai_eg
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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

−8.107212734054336291985733923038, −7.910050612817412619434815276765, −7.42663021842369398843361676359, −7.15438881941100593242563888383, −6.47538037687462323460881315962, −6.07454615634188289333576083777, −5.44159911512899854367102901077, −4.81290415692609065796084392684, −4.43879684292035116742208075197, −3.63140428142550758563754653634, −3.17211385217539487216733044205, −2.40967405882884571062551300590, −1.88464287650654210018176627605, −1.14238250962966379214471971375, 0, 1.14238250962966379214471971375, 1.88464287650654210018176627605, 2.40967405882884571062551300590, 3.17211385217539487216733044205, 3.63140428142550758563754653634, 4.43879684292035116742208075197, 4.81290415692609065796084392684, 5.44159911512899854367102901077, 6.07454615634188289333576083777, 6.47538037687462323460881315962, 7.15438881941100593242563888383, 7.42663021842369398843361676359, 7.910050612817412619434815276765, 8.107212734054336291985733923038

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