Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 167 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3.83·5-s − 4.48·7-s + 9-s − 1.10·11-s − 5.67·13-s + 3.83·15-s − 0.581·17-s + 4.72·19-s − 4.48·21-s + 3.33·23-s + 9.72·25-s + 27-s + 3.38·29-s − 2.78·31-s − 1.10·33-s − 17.1·35-s + 4.43·37-s − 5.67·39-s + 5.90·41-s − 10.0·43-s + 3.83·45-s + 12.4·47-s + 13.0·49-s − 0.581·51-s + 5.10·53-s − 4.24·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.71·5-s − 1.69·7-s + 0.333·9-s − 0.333·11-s − 1.57·13-s + 0.990·15-s − 0.140·17-s + 1.08·19-s − 0.978·21-s + 0.695·23-s + 1.94·25-s + 0.192·27-s + 0.628·29-s − 0.499·31-s − 0.192·33-s − 2.90·35-s + 0.728·37-s − 0.908·39-s + 0.921·41-s − 1.52·43-s + 0.571·45-s + 1.80·47-s + 1.86·49-s − 0.0813·51-s + 0.701·53-s − 0.572·55-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  =  \(0\)
Selberg data  =  \((2,\ 8016,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.728013025\)
\(L(\frac12)\)  \(\approx\)  \(2.728013025\)
\(L(\frac{3}{2})\)   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 \( 1 - 3.83T + 5T^{2} \)
7 \( 1 + 4.48T + 7T^{2} \)
11 \( 1 + 1.10T + 11T^{2} \)
13 \( 1 + 5.67T + 13T^{2} \)
17 \( 1 + 0.581T + 17T^{2} \)
19 \( 1 - 4.72T + 19T^{2} \)
23 \( 1 - 3.33T + 23T^{2} \)
29 \( 1 - 3.38T + 29T^{2} \)
31 \( 1 + 2.78T + 31T^{2} \)
37 \( 1 - 4.43T + 37T^{2} \)
41 \( 1 - 5.90T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 - 5.10T + 53T^{2} \)
59 \( 1 - 2.77T + 59T^{2} \)
61 \( 1 - 9.42T + 61T^{2} \)
67 \( 1 + 9.07T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 9.14T + 73T^{2} \)
79 \( 1 - 17.4T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + 1.50T + 89T^{2} \)
97 \( 1 - 0.761T + 97T^{2} \)
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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

−7.66545906894652989091494023944, −7.00310536776741136957016519745, −6.58102193417165106262428402016, −5.66109831683722380487480967751, −5.29707907966295399733398889226, −4.28286542923298333044922799299, −3.07640677691538403117644255164, −2.75847918444987791047064168165, −2.05208593552907738799192087809, −0.77070989295029803958593360929, 0.77070989295029803958593360929, 2.05208593552907738799192087809, 2.75847918444987791047064168165, 3.07640677691538403117644255164, 4.28286542923298333044922799299, 5.29707907966295399733398889226, 5.66109831683722380487480967751, 6.58102193417165106262428402016, 7.00310536776741136957016519745, 7.66545906894652989091494023944

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