Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 167 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 0.614·5-s + 3.46·7-s + 9-s − 3.80·11-s − 2.93·13-s + 0.614·15-s + 3.15·17-s + 5.58·19-s − 3.46·21-s + 0.574·23-s − 4.62·25-s − 27-s − 3.25·29-s + 1.74·31-s + 3.80·33-s − 2.12·35-s − 4.35·37-s + 2.93·39-s − 1.51·41-s − 9.97·43-s − 0.614·45-s + 8.29·47-s + 4.99·49-s − 3.15·51-s + 0.465·53-s + 2.34·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.275·5-s + 1.30·7-s + 0.333·9-s − 1.14·11-s − 0.812·13-s + 0.158·15-s + 0.765·17-s + 1.28·19-s − 0.755·21-s + 0.119·23-s − 0.924·25-s − 0.192·27-s − 0.603·29-s + 0.313·31-s + 0.663·33-s − 0.359·35-s − 0.716·37-s + 0.469·39-s − 0.236·41-s − 1.52·43-s − 0.0916·45-s + 1.20·47-s + 0.713·49-s − 0.441·51-s + 0.0638·53-s + 0.315·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8016} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 8016,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
167 \( 1 - T \)
good5 \( 1 + 0.614T + 5T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 + 3.80T + 11T^{2} \)
13 \( 1 + 2.93T + 13T^{2} \)
17 \( 1 - 3.15T + 17T^{2} \)
19 \( 1 - 5.58T + 19T^{2} \)
23 \( 1 - 0.574T + 23T^{2} \)
29 \( 1 + 3.25T + 29T^{2} \)
31 \( 1 - 1.74T + 31T^{2} \)
37 \( 1 + 4.35T + 37T^{2} \)
41 \( 1 + 1.51T + 41T^{2} \)
43 \( 1 + 9.97T + 43T^{2} \)
47 \( 1 - 8.29T + 47T^{2} \)
53 \( 1 - 0.465T + 53T^{2} \)
59 \( 1 - 9.58T + 59T^{2} \)
61 \( 1 + 6.22T + 61T^{2} \)
67 \( 1 + 3.41T + 67T^{2} \)
71 \( 1 - 8.58T + 71T^{2} \)
73 \( 1 + 9.90T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 - 8.39T + 83T^{2} \)
89 \( 1 + 7.12T + 89T^{2} \)
97 \( 1 - 4.70T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.57500697412935966083318325002, −7.03159731142673937101313929924, −5.84766499767628997238390920654, −5.22192149346962619483050674292, −4.98489599754750211636755983909, −4.02933269567062350680807549684, −3.09441225749019886403543995806, −2.12362682509620663223155997689, −1.22707057726017983380740648444, 0, 1.22707057726017983380740648444, 2.12362682509620663223155997689, 3.09441225749019886403543995806, 4.02933269567062350680807549684, 4.98489599754750211636755983909, 5.22192149346962619483050674292, 5.84766499767628997238390920654, 7.03159731142673937101313929924, 7.57500697412935966083318325002

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