Properties

Label 4-546e2-1.1-c1e2-0-58
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 3·4-s + 6·6-s − 5·7-s + 4·8-s + 6·9-s + 9·12-s + 2·13-s − 10·14-s + 5·16-s + 6·17-s + 12·18-s + 4·19-s − 15·21-s + 12·24-s + 7·25-s + 4·26-s + 9·27-s − 15·28-s − 16·31-s + 6·32-s + 12·34-s + 18·36-s + 8·38-s + 6·39-s − 30·42-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 3/2·4-s + 2.44·6-s − 1.88·7-s + 1.41·8-s + 2·9-s + 2.59·12-s + 0.554·13-s − 2.67·14-s + 5/4·16-s + 1.45·17-s + 2.82·18-s + 0.917·19-s − 3.27·21-s + 2.44·24-s + 7/5·25-s + 0.784·26-s + 1.73·27-s − 2.83·28-s − 2.87·31-s + 1.06·32-s + 2.05·34-s + 3·36-s + 1.29·38-s + 0.960·39-s − 4.62·42-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: induced by $\chi_{546} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.388204668\)
\(L(\frac12)\) \(\approx\) \(7.388204668\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 166 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−10.84509625128243547410197959699, −10.64699407848347568329041107502, −10.06353085932936194211243375785, −9.593923597260431743651513723146, −9.389059942041548515016903396601, −8.905848056954869280083780155421, −8.224239765567810305698716476228, −7.912272905360040515456890246534, −7.12696457284613766095951818614, −7.07115744253702890159614085922, −6.53349746314984434834244872719, −5.97424731968802049408636226979, −5.21182945760726672695674910454, −5.09740903242519650302606083236, −3.90665101802109905137323686306, −3.58850534674695920434636068960, −3.35154370495190499162848631191, −2.98194318004286500581533834895, −2.20195601855288858272551951258, −1.34187722864537857370977683860, 1.34187722864537857370977683860, 2.20195601855288858272551951258, 2.98194318004286500581533834895, 3.35154370495190499162848631191, 3.58850534674695920434636068960, 3.90665101802109905137323686306, 5.09740903242519650302606083236, 5.21182945760726672695674910454, 5.97424731968802049408636226979, 6.53349746314984434834244872719, 7.07115744253702890159614085922, 7.12696457284613766095951818614, 7.912272905360040515456890246534, 8.224239765567810305698716476228, 8.905848056954869280083780155421, 9.389059942041548515016903396601, 9.593923597260431743651513723146, 10.06353085932936194211243375785, 10.64699407848347568329041107502, 10.84509625128243547410197959699

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