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 + 0.0621·5-s − 0.782·7-s + 9-s − 2.68·11-s + 6.79·13-s + 0.0621·15-s + 4.20·17-s − 7.94·19-s − 0.782·21-s − 5.42·23-s − 4.99·25-s + 27-s + 1.95·29-s − 2.86·31-s − 2.68·33-s − 0.0486·35-s − 7.68·37-s + 6.79·39-s − 2.03·41-s + 5.17·43-s + 0.0621·45-s + 7.62·47-s − 6.38·49-s + 4.20·51-s + 3.26·53-s − 0.166·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.0278·5-s − 0.295·7-s + 0.333·9-s − 0.809·11-s + 1.88·13-s + 0.0160·15-s + 1.02·17-s − 1.82·19-s − 0.170·21-s − 1.13·23-s − 0.999·25-s + 0.192·27-s + 0.363·29-s − 0.514·31-s − 0.467·33-s − 0.00822·35-s − 1.26·37-s + 1.08·39-s − 0.317·41-s + 0.788·43-s + 0.00926·45-s + 1.11·47-s − 0.912·49-s + 0.589·51-s + 0.448·53-s − 0.0225·55-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 \( 1 - 0.0621T + 5T^{2} \)
7 \( 1 + 0.782T + 7T^{2} \)
11 \( 1 + 2.68T + 11T^{2} \)
13 \( 1 - 6.79T + 13T^{2} \)
17 \( 1 - 4.20T + 17T^{2} \)
19 \( 1 + 7.94T + 19T^{2} \)
23 \( 1 + 5.42T + 23T^{2} \)
29 \( 1 - 1.95T + 29T^{2} \)
31 \( 1 + 2.86T + 31T^{2} \)
37 \( 1 + 7.68T + 37T^{2} \)
41 \( 1 + 2.03T + 41T^{2} \)
43 \( 1 - 5.17T + 43T^{2} \)
47 \( 1 - 7.62T + 47T^{2} \)
53 \( 1 - 3.26T + 53T^{2} \)
59 \( 1 + 2.33T + 59T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 - 1.25T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + 3.71T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 2.74T + 89T^{2} \)
97 \( 1 - 1.54T + 97T^{2} \)
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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

−7.69375874121797739412553314056, −6.77939264875556550787581886064, −5.99485282259042052209253286105, −5.66412468911587774822353352511, −4.43197894808133002894803481947, −3.82469814929521775788792840879, −3.20261921964188226402211756420, −2.20712594402098554688672285988, −1.43864294588264191294012905043, 0, 1.43864294588264191294012905043, 2.20712594402098554688672285988, 3.20261921964188226402211756420, 3.82469814929521775788792840879, 4.43197894808133002894803481947, 5.66412468911587774822353352511, 5.99485282259042052209253286105, 6.77939264875556550787581886064, 7.69375874121797739412553314056

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