Properties

Label 2-8016-1.1-c1-0-30
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 − 3.16·5-s + 0.230·7-s + 9-s − 0.534·11-s + 2.40·13-s + 3.16·15-s + 3.83·17-s + 7.74·19-s − 0.230·21-s − 5.14·23-s + 5.03·25-s − 27-s + 0.602·29-s + 4.43·31-s + 0.534·33-s − 0.731·35-s + 8.52·37-s − 2.40·39-s + 1.40·41-s + 3.66·43-s − 3.16·45-s − 12.4·47-s − 6.94·49-s − 3.83·51-s + 4.85·53-s + 1.69·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.41·5-s + 0.0872·7-s + 0.333·9-s − 0.161·11-s + 0.666·13-s + 0.818·15-s + 0.930·17-s + 1.77·19-s − 0.0503·21-s − 1.07·23-s + 1.00·25-s − 0.192·27-s + 0.111·29-s + 0.797·31-s + 0.0931·33-s − 0.123·35-s + 1.40·37-s − 0.385·39-s + 0.219·41-s + 0.559·43-s − 0.472·45-s − 1.82·47-s − 0.992·49-s − 0.537·51-s + 0.667·53-s + 0.228·55-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

\(L(1)\) \(\approx\) \(1.183110921\)
\(L(\frac12)\) \(\approx\) \(1.183110921\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.16T + 5T^{2} \)
7 \( 1 - 0.230T + 7T^{2} \)
11 \( 1 + 0.534T + 11T^{2} \)
13 \( 1 - 2.40T + 13T^{2} \)
17 \( 1 - 3.83T + 17T^{2} \)
19 \( 1 - 7.74T + 19T^{2} \)
23 \( 1 + 5.14T + 23T^{2} \)
29 \( 1 - 0.602T + 29T^{2} \)
31 \( 1 - 4.43T + 31T^{2} \)
37 \( 1 - 8.52T + 37T^{2} \)
41 \( 1 - 1.40T + 41T^{2} \)
43 \( 1 - 3.66T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 - 4.85T + 53T^{2} \)
59 \( 1 + 9.66T + 59T^{2} \)
61 \( 1 + 1.49T + 61T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 - 4.16T + 73T^{2} \)
79 \( 1 - 7.52T + 79T^{2} \)
83 \( 1 + 7.44T + 83T^{2} \)
89 \( 1 - 9.40T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−7.76635635863299992339458345472, −7.40264152663654323534088565115, −6.35142223930754145665138623692, −5.82931745406269409263515013972, −4.92772342656073502964882323805, −4.33982614891683422272725227528, −3.51663286854926027053246109800, −2.98003329983315880428754068336, −1.46059148731946406086101808548, −0.60014512933024843887347739797, 0.60014512933024843887347739797, 1.46059148731946406086101808548, 2.98003329983315880428754068336, 3.51663286854926027053246109800, 4.33982614891683422272725227528, 4.92772342656073502964882323805, 5.82931745406269409263515013972, 6.35142223930754145665138623692, 7.40264152663654323534088565115, 7.76635635863299992339458345472

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