Properties

Label 2-8016-1.1-c1-0-22
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 − 1.69·5-s + 4.12·7-s + 9-s − 2.48·11-s + 1.35·13-s + 1.69·15-s − 7.89·17-s − 5.08·19-s − 4.12·21-s − 6.55·23-s − 2.13·25-s − 27-s + 7.41·29-s − 4.69·31-s + 2.48·33-s − 6.99·35-s − 0.0133·37-s − 1.35·39-s − 1.02·41-s + 8.67·43-s − 1.69·45-s − 5.31·47-s + 10.0·49-s + 7.89·51-s + 12.1·53-s + 4.21·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.757·5-s + 1.56·7-s + 0.333·9-s − 0.750·11-s + 0.375·13-s + 0.437·15-s − 1.91·17-s − 1.16·19-s − 0.901·21-s − 1.36·23-s − 0.426·25-s − 0.192·27-s + 1.37·29-s − 0.843·31-s + 0.433·33-s − 1.18·35-s − 0.00219·37-s − 0.216·39-s − 0.159·41-s + 1.32·43-s − 0.252·45-s − 0.775·47-s + 1.43·49-s + 1.10·51-s + 1.66·53-s + 0.568·55-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

\(L(1)\) \(\approx\) \(1.033047523\)
\(L(\frac12)\) \(\approx\) \(1.033047523\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.69T + 5T^{2} \)
7 \( 1 - 4.12T + 7T^{2} \)
11 \( 1 + 2.48T + 11T^{2} \)
13 \( 1 - 1.35T + 13T^{2} \)
17 \( 1 + 7.89T + 17T^{2} \)
19 \( 1 + 5.08T + 19T^{2} \)
23 \( 1 + 6.55T + 23T^{2} \)
29 \( 1 - 7.41T + 29T^{2} \)
31 \( 1 + 4.69T + 31T^{2} \)
37 \( 1 + 0.0133T + 37T^{2} \)
41 \( 1 + 1.02T + 41T^{2} \)
43 \( 1 - 8.67T + 43T^{2} \)
47 \( 1 + 5.31T + 47T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 - 0.0660T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 6.10T + 67T^{2} \)
71 \( 1 - 9.95T + 71T^{2} \)
73 \( 1 + 3.76T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 - 6.46T + 89T^{2} \)
97 \( 1 - 18.7T + 97T^{2} \)
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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

−7.82394299746994349413179148918, −7.25969345691189981909150164570, −6.41044595680615961224769084126, −5.74726457991686658008574792913, −4.86215783199709304634259909814, −4.37090855380444991866626157130, −3.88300399844791519322632205128, −2.38089769095761072093295443866, −1.86698230614978808121087263497, −0.50421588409354719836052256645, 0.50421588409354719836052256645, 1.86698230614978808121087263497, 2.38089769095761072093295443866, 3.88300399844791519322632205128, 4.37090855380444991866626157130, 4.86215783199709304634259909814, 5.74726457991686658008574792913, 6.41044595680615961224769084126, 7.25969345691189981909150164570, 7.82394299746994349413179148918

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