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Properties

Label 4-1812608-1.1-c1e2-0-7
Degree $4$
Conductor $1812608$
Sign $-1$
Analytic cond. $115.573$
Root an. cond. $3.27879$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 6·9-s + 16-s + 2·17-s − 6·18-s + 8·19-s − 6·25-s + 32-s + 2·34-s − 6·36-s + 8·38-s − 12·41-s − 24·43-s + 49-s − 6·50-s + 8·59-s + 64-s + 24·67-s + 2·68-s − 6·72-s + 4·73-s + 8·76-s + 27·81-s − 12·82-s + 24·83-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2·9-s + 1/4·16-s + 0.485·17-s − 1.41·18-s + 1.83·19-s − 6/5·25-s + 0.176·32-s + 0.342·34-s − 36-s + 1.29·38-s − 1.87·41-s − 3.65·43-s + 1/7·49-s − 0.848·50-s + 1.04·59-s + 1/8·64-s + 2.93·67-s + 0.242·68-s − 0.707·72-s + 0.468·73-s + 0.917·76-s + 3·81-s − 1.32·82-s + 2.63·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1812608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1812608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1812608\)    =    \(2^{7} \cdot 7^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(115.573\)
Root analytic conductor: \(3.27879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1812608,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.43.y_iw
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−7.71282024817693675745013733529, −6.94219495068550059915873452189, −6.73763985186732377586137626470, −6.26990474233050369942971608193, −5.70286626752820956259075840901, −5.35539917644036382158361461516, −5.11332775341498038219899119581, −4.77682145642589218436996001225, −3.63026567116244883219000059570, −3.40674497567057131140915818833, −3.36366542167825731325973439863, −2.40395727895840686691357326236, −2.04650857588819893440526220286, −1.08252162774259574666896222029, 0, 1.08252162774259574666896222029, 2.04650857588819893440526220286, 2.40395727895840686691357326236, 3.36366542167825731325973439863, 3.40674497567057131140915818833, 3.63026567116244883219000059570, 4.77682145642589218436996001225, 5.11332775341498038219899119581, 5.35539917644036382158361461516, 5.70286626752820956259075840901, 6.26990474233050369942971608193, 6.73763985186732377586137626470, 6.94219495068550059915873452189, 7.71282024817693675745013733529

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