Properties

Label 2-8016-1.1-c1-0-148
Degree $2$
Conductor $8016$
Sign $-1$
Analytic cond. $64.0080$
Root an. cond. $8.00050$
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  + 3-s + 3·5-s − 3·7-s + 9-s − 6·11-s + 2·13-s + 3·15-s − 3·21-s − 4·23-s + 4·25-s + 27-s − 6·29-s + 9·31-s − 6·33-s − 9·35-s − 37-s + 2·39-s + 2·41-s + 8·43-s + 3·45-s + 47-s + 2·49-s − 3·53-s − 18·55-s − 3·59-s − 12·61-s − 3·63-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s − 1.13·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 0.774·15-s − 0.654·21-s − 0.834·23-s + 4/5·25-s + 0.192·27-s − 1.11·29-s + 1.61·31-s − 1.04·33-s − 1.52·35-s − 0.164·37-s + 0.320·39-s + 0.312·41-s + 1.21·43-s + 0.447·45-s + 0.145·47-s + 2/7·49-s − 0.412·53-s − 2.42·55-s − 0.390·59-s − 1.53·61-s − 0.377·63-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: yes
Arithmetic: yes
Character: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 + T \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−7.59243473359626210249280313012, −6.70122670741968090403038283180, −5.98012258068665926734658390396, −5.65667118127924341008744784578, −4.71547918020380636961303342443, −3.75559621001915142286183865747, −2.77962630596731654963452803345, −2.51293704954669834899667068711, −1.45851962934809435692307859296, 0, 1.45851962934809435692307859296, 2.51293704954669834899667068711, 2.77962630596731654963452803345, 3.75559621001915142286183865747, 4.71547918020380636961303342443, 5.65667118127924341008744784578, 5.98012258068665926734658390396, 6.70122670741968090403038283180, 7.59243473359626210249280313012

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