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Properties

Label 4-546e2-1.1-c1e2-0-14
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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  − 3-s − 4-s − 2·9-s + 12-s + 16-s + 11·17-s + 6·25-s + 5·27-s − 9·29-s + 2·36-s − 43-s − 48-s − 49-s − 11·51-s − 5·53-s + 13·61-s − 64-s − 11·68-s − 6·75-s − 15·79-s + 81-s + 9·87-s − 6·100-s + 11·101-s − 103-s + 26·107-s − 5·108-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s − 2/3·9-s + 0.288·12-s + 1/4·16-s + 2.66·17-s + 6/5·25-s + 0.962·27-s − 1.67·29-s + 1/3·36-s − 0.152·43-s − 0.144·48-s − 1/7·49-s − 1.54·51-s − 0.686·53-s + 1.66·61-s − 1/8·64-s − 1.33·68-s − 0.692·75-s − 1.68·79-s + 1/9·81-s + 0.964·87-s − 3/5·100-s + 1.09·101-s − 0.0985·103-s + 2.51·107-s − 0.481·108-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.147768520\)
\(L(\frac12)\) \(\approx\) \(1.147768520\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.11.a_d
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.17.al_ck
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.19.a_u
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.j_cy
31$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.31.a_ae
37$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.37.a_h
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.41.a_be
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.b_co
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.a_ag
53$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.53.f_c
59$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \) 2.59.a_cb
61$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.an_fo
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.71.a_bv
73$C_2^2$ \( 1 - 115 T^{2} + p^{2} T^{4} \) 2.73.a_ael
79$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.79.p_ie
83$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.83.a_adq
89$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.89.a_cw
97$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.97.a_f
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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

−8.801865297960571300465901701440, −8.355259144899635669374028917046, −7.977495753198467776695010790032, −7.34421753504590671509703779979, −7.13950789800159507342557907464, −6.23289191305093910306445489370, −5.94025866236808335791283345861, −5.31115087019525329111918769984, −5.22265996614244128825894790351, −4.50681747145871408585541457793, −3.65705823831213042471955558700, −3.34349869514538042531219722257, −2.67985472529534111917542756516, −1.55844681266113046809237210397, −0.69212136137643190815272938222, 0.69212136137643190815272938222, 1.55844681266113046809237210397, 2.67985472529534111917542756516, 3.34349869514538042531219722257, 3.65705823831213042471955558700, 4.50681747145871408585541457793, 5.22265996614244128825894790351, 5.31115087019525329111918769984, 5.94025866236808335791283345861, 6.23289191305093910306445489370, 7.13950789800159507342557907464, 7.34421753504590671509703779979, 7.977495753198467776695010790032, 8.355259144899635669374028917046, 8.801865297960571300465901701440

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