Properties

Label 2-8016-1.1-c1-0-0
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 − 0.792·5-s − 3.80·7-s + 9-s − 5.89·11-s − 0.555·13-s + 0.792·15-s − 1.29·17-s + 0.877·19-s + 3.80·21-s − 1.89·23-s − 4.37·25-s − 27-s + 4.21·29-s − 6.06·31-s + 5.89·33-s + 3.01·35-s − 8.81·37-s + 0.555·39-s + 0.966·41-s − 1.48·43-s − 0.792·45-s − 4.12·47-s + 7.45·49-s + 1.29·51-s − 13.1·53-s + 4.66·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.354·5-s − 1.43·7-s + 0.333·9-s − 1.77·11-s − 0.154·13-s + 0.204·15-s − 0.313·17-s + 0.201·19-s + 0.829·21-s − 0.394·23-s − 0.874·25-s − 0.192·27-s + 0.782·29-s − 1.08·31-s + 1.02·33-s + 0.509·35-s − 1.44·37-s + 0.0889·39-s + 0.151·41-s − 0.226·43-s − 0.118·45-s − 0.601·47-s + 1.06·49-s + 0.180·51-s − 1.80·53-s + 0.629·55-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

\(L(1)\) \(\approx\) \(0.01373216811\)
\(L(\frac12)\) \(\approx\) \(0.01373216811\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.792T + 5T^{2} \)
7 \( 1 + 3.80T + 7T^{2} \)
11 \( 1 + 5.89T + 11T^{2} \)
13 \( 1 + 0.555T + 13T^{2} \)
17 \( 1 + 1.29T + 17T^{2} \)
19 \( 1 - 0.877T + 19T^{2} \)
23 \( 1 + 1.89T + 23T^{2} \)
29 \( 1 - 4.21T + 29T^{2} \)
31 \( 1 + 6.06T + 31T^{2} \)
37 \( 1 + 8.81T + 37T^{2} \)
41 \( 1 - 0.966T + 41T^{2} \)
43 \( 1 + 1.48T + 43T^{2} \)
47 \( 1 + 4.12T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 7.38T + 61T^{2} \)
67 \( 1 - 6.38T + 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 0.414T + 79T^{2} \)
83 \( 1 - 2.57T + 83T^{2} \)
89 \( 1 - 0.384T + 89T^{2} \)
97 \( 1 + 3.26T + 97T^{2} \)
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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

−7.70786910416393214034033140748, −7.15309527415192395109497991776, −6.39130259146502201083249221025, −5.79343598945988638634619888604, −5.12091686664944850534868966191, −4.36288714821197665838047027225, −3.37007085915841495906993131674, −2.87031154717512854089266876593, −1.77357158003425738843707968826, −0.05609283255498602513243611204, 0.05609283255498602513243611204, 1.77357158003425738843707968826, 2.87031154717512854089266876593, 3.37007085915841495906993131674, 4.36288714821197665838047027225, 5.12091686664944850534868966191, 5.79343598945988638634619888604, 6.39130259146502201083249221025, 7.15309527415192395109497991776, 7.70786910416393214034033140748

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