Properties

Label 2-8016-1.1-c1-0-41
Degree $2$
Conductor $8016$
Sign $1$
Analytic cond. $64.0080$
Root an. cond. $8.00050$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 1.90·5-s − 2.81·7-s + 9-s − 0.318·11-s + 5.12·13-s − 1.90·15-s + 3.73·17-s + 0.725·19-s + 2.81·21-s − 0.612·23-s − 1.36·25-s − 27-s − 3.87·29-s + 6.65·31-s + 0.318·33-s − 5.36·35-s − 9.04·37-s − 5.12·39-s + 9.55·41-s + 10.9·43-s + 1.90·45-s − 8.76·47-s + 0.915·49-s − 3.73·51-s − 2.46·53-s − 0.608·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.853·5-s − 1.06·7-s + 0.333·9-s − 0.0961·11-s + 1.42·13-s − 0.492·15-s + 0.905·17-s + 0.166·19-s + 0.613·21-s − 0.127·23-s − 0.272·25-s − 0.192·27-s − 0.720·29-s + 1.19·31-s + 0.0555·33-s − 0.907·35-s − 1.48·37-s − 0.820·39-s + 1.49·41-s + 1.66·43-s + 0.284·45-s − 1.27·47-s + 0.130·49-s − 0.522·51-s − 0.338·53-s − 0.0820·55-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

Degree: \(2\)
Conductor: \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
Sign: $1$
Analytic conductor: \(64.0080\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.847630146\)
\(L(\frac12)\) \(\approx\) \(1.847630146\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
167 \( 1 + T \)
good5 \( 1 - 1.90T + 5T^{2} \)
7 \( 1 + 2.81T + 7T^{2} \)
11 \( 1 + 0.318T + 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 - 3.73T + 17T^{2} \)
19 \( 1 - 0.725T + 19T^{2} \)
23 \( 1 + 0.612T + 23T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 - 6.65T + 31T^{2} \)
37 \( 1 + 9.04T + 37T^{2} \)
41 \( 1 - 9.55T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + 8.76T + 47T^{2} \)
53 \( 1 + 2.46T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + 5.21T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 2.88T + 73T^{2} \)
79 \( 1 + 8.98T + 79T^{2} \)
83 \( 1 + 4.04T + 83T^{2} \)
89 \( 1 + 9.15T + 89T^{2} \)
97 \( 1 + 9.62T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.81356275660596879439209373733, −6.87129079012052857116102238418, −6.39263923628762117790437488970, −5.69273431444510670879876141398, −5.44631713898169445502898796202, −4.18537634409050719835908481761, −3.55333512821485734010947965386, −2.70607886251700332757295447270, −1.63502182877443115419705333111, −0.71984961285225384124660914258, 0.71984961285225384124660914258, 1.63502182877443115419705333111, 2.70607886251700332757295447270, 3.55333512821485734010947965386, 4.18537634409050719835908481761, 5.44631713898169445502898796202, 5.69273431444510670879876141398, 6.39263923628762117790437488970, 6.87129079012052857116102238418, 7.81356275660596879439209373733

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