Properties

Label 4-642816-1.1-c1e2-0-6
Degree $4$
Conductor $642816$
Sign $-1$
Analytic cond. $40.9865$
Root an. cond. $2.53023$
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  − 5·7-s + 7·13-s − 5·19-s − 25-s + 6·31-s + 37-s − 10·43-s + 5·49-s − 61-s + 10·67-s + 7·73-s − 5·79-s − 35·91-s − 6·97-s − 15·103-s − 5·109-s − 3·121-s + 127-s + 131-s + 25·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.88·7-s + 1.94·13-s − 1.14·19-s − 1/5·25-s + 1.07·31-s + 0.164·37-s − 1.52·43-s + 5/7·49-s − 0.128·61-s + 1.22·67-s + 0.819·73-s − 0.562·79-s − 3.66·91-s − 0.609·97-s − 1.47·103-s − 0.478·109-s − 0.272·121-s + 0.0887·127-s + 0.0873·131-s + 2.16·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 642816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 642816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(642816\)    =    \(2^{8} \cdot 3^{4} \cdot 31\)
Sign: $-1$
Analytic conductor: \(40.9865\)
Root analytic conductor: \(2.53023\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{642816} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 642816,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
31$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 7 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−8.317621787385298507202012330783, −7.85005569835851696135490186992, −6.96976096601152499231061653242, −6.66513851093469571005050562382, −6.39912503789148775338339206514, −6.05086180025857356810690580320, −5.54740182790996972982760517119, −4.86954230756168018019880850835, −4.15389357184755007458811156734, −3.78981354795991226802822633972, −3.27679983906830999819341675998, −2.85197644295622489506873608302, −2.00899033612871943378369886414, −1.10808742039124466181003510318, 0, 1.10808742039124466181003510318, 2.00899033612871943378369886414, 2.85197644295622489506873608302, 3.27679983906830999819341675998, 3.78981354795991226802822633972, 4.15389357184755007458811156734, 4.86954230756168018019880850835, 5.54740182790996972982760517119, 6.05086180025857356810690580320, 6.39912503789148775338339206514, 6.66513851093469571005050562382, 6.96976096601152499231061653242, 7.85005569835851696135490186992, 8.317621787385298507202012330783

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