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Properties

Label 4-546e2-1.1-c1e2-0-53
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·7-s − 3·9-s + 16-s − 6·25-s − 2·28-s + 3·36-s − 12·37-s + 16·43-s − 3·49-s − 6·63-s − 64-s − 4·67-s − 8·79-s + 9·81-s + 6·100-s − 20·109-s + 2·112-s + 6·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 12·148-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.755·7-s − 9-s + 1/4·16-s − 6/5·25-s − 0.377·28-s + 1/2·36-s − 1.97·37-s + 2.43·43-s − 3/7·49-s − 0.755·63-s − 1/8·64-s − 0.488·67-s − 0.900·79-s + 81-s + 3/5·100-s − 1.91·109-s + 0.188·112-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + 0.986·148-s + 0.0819·149-s + 0.0813·151-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: \(1\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.71.a_ek
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.a_aby
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.a_dq
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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

−8.612050380929465536294756200632, −8.192457492943088357975137948619, −7.71552888273325185252964381749, −7.37843150622851074079273422611, −6.70486662775792515336485617966, −6.05452355793855226498498579130, −5.70105711438140951172877746729, −5.20373884652850839570419096346, −4.75006012808895696131814676563, −4.04444528003798932299298117492, −3.64449768367960580971342056499, −2.83900758560045791802133984165, −2.19475977897524993954027993452, −1.32745879898891919092848453705, 0, 1.32745879898891919092848453705, 2.19475977897524993954027993452, 2.83900758560045791802133984165, 3.64449768367960580971342056499, 4.04444528003798932299298117492, 4.75006012808895696131814676563, 5.20373884652850839570419096346, 5.70105711438140951172877746729, 6.05452355793855226498498579130, 6.70486662775792515336485617966, 7.37843150622851074079273422611, 7.71552888273325185252964381749, 8.192457492943088357975137948619, 8.612050380929465536294756200632

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