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Properties

Label 4-588e2-1.1-c1e2-0-31
Degree $4$
Conductor $345744$
Sign $-1$
Analytic cond. $22.0449$
Root an. cond. $2.16684$
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 − 4-s − 3·8-s − 9-s + 2·11-s − 16-s − 18-s + 2·22-s + 2·23-s − 4·25-s − 8·29-s + 5·32-s + 36-s − 2·44-s + 2·46-s − 4·50-s + 8·53-s − 8·58-s + 7·64-s − 12·67-s − 18·71-s + 3·72-s − 4·79-s + 81-s − 6·88-s − 2·92-s − 2·99-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1/3·9-s + 0.603·11-s − 1/4·16-s − 0.235·18-s + 0.426·22-s + 0.417·23-s − 4/5·25-s − 1.48·29-s + 0.883·32-s + 1/6·36-s − 0.301·44-s + 0.294·46-s − 0.565·50-s + 1.09·53-s − 1.05·58-s + 7/8·64-s − 1.46·67-s − 2.13·71-s + 0.353·72-s − 0.450·79-s + 1/9·81-s − 0.639·88-s − 0.208·92-s − 0.201·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(345744\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(22.0449\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 345744,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.ac_o
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.13.a_i
17$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.17.a_u
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.19.a_o
23$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ac_bu
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.i_cs
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.a_abm
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.41.a_au
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.ai_w
59$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.59.a_o
61$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.61.a_q
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.m_cs
71$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.s_io
73$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.73.a_abo
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.e_ew
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.83.a_ba
89$C_2^2$ \( 1 + 156 T^{2} + p^{2} T^{4} \) 2.89.a_ga
97$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.97.a_bo
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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

−8.739669822998969445809140863235, −8.034516887502443310649335442709, −7.52685283860693278740387088025, −7.15143906922658593640837830289, −6.45585929004488164739050655816, −5.99444197952005469662508385604, −5.64425540196005984867157019856, −5.16050516489234500565730154860, −4.52358344101759891331149131012, −4.06963482182445630773446579280, −3.58887616485273778765261206279, −3.02437764916907353123409236856, −2.28485132977309640466176859544, −1.33358386294238772488720373779, 0, 1.33358386294238772488720373779, 2.28485132977309640466176859544, 3.02437764916907353123409236856, 3.58887616485273778765261206279, 4.06963482182445630773446579280, 4.52358344101759891331149131012, 5.16050516489234500565730154860, 5.64425540196005984867157019856, 5.99444197952005469662508385604, 6.45585929004488164739050655816, 7.15143906922658593640837830289, 7.52685283860693278740387088025, 8.034516887502443310649335442709, 8.739669822998969445809140863235

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