Properties

Degree $2$
Conductor $8016$
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 − 1.68·5-s + 2.23·7-s + 9-s + 3.03·11-s − 4.55·13-s − 1.68·15-s − 2.98·17-s + 2.14·19-s + 2.23·21-s − 1.24·23-s − 2.16·25-s + 27-s − 6.48·29-s − 0.905·31-s + 3.03·33-s − 3.75·35-s + 5.19·37-s − 4.55·39-s + 0.662·41-s + 10.7·43-s − 1.68·45-s − 6.58·47-s − 2.02·49-s − 2.98·51-s − 8.56·53-s − 5.11·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.753·5-s + 0.843·7-s + 0.333·9-s + 0.914·11-s − 1.26·13-s − 0.435·15-s − 0.724·17-s + 0.493·19-s + 0.486·21-s − 0.259·23-s − 0.432·25-s + 0.192·27-s − 1.20·29-s − 0.162·31-s + 0.528·33-s − 0.635·35-s + 0.853·37-s − 0.729·39-s + 0.103·41-s + 1.63·43-s − 0.251·45-s − 0.960·47-s − 0.289·49-s − 0.418·51-s − 1.17·53-s − 0.689·55-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: $-1$
Motivic weight: \(1\)
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 \( 1 + 1.68T + 5T^{2} \)
7 \( 1 - 2.23T + 7T^{2} \)
11 \( 1 - 3.03T + 11T^{2} \)
13 \( 1 + 4.55T + 13T^{2} \)
17 \( 1 + 2.98T + 17T^{2} \)
19 \( 1 - 2.14T + 19T^{2} \)
23 \( 1 + 1.24T + 23T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 + 0.905T + 31T^{2} \)
37 \( 1 - 5.19T + 37T^{2} \)
41 \( 1 - 0.662T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + 6.58T + 47T^{2} \)
53 \( 1 + 8.56T + 53T^{2} \)
59 \( 1 - 0.576T + 59T^{2} \)
61 \( 1 + 9.76T + 61T^{2} \)
67 \( 1 + 4.66T + 67T^{2} \)
71 \( 1 + 2.31T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 + 6.77T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + 1.55T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−7.71588338740422978897861144356, −7.03214294269886486557587419388, −6.18148720678301011975289476761, −5.24527273422373753206527472526, −4.43313132356393948552380725134, −4.06346653804421548965042447360, −3.11682998625577729140061692263, −2.21243375175487858723331149449, −1.41001037265649908373912207860, 0, 1.41001037265649908373912207860, 2.21243375175487858723331149449, 3.11682998625577729140061692263, 4.06346653804421548965042447360, 4.43313132356393948552380725134, 5.24527273422373753206527472526, 6.18148720678301011975289476761, 7.03214294269886486557587419388, 7.71588338740422978897861144356

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