Properties

Degree $2$
Conductor $8034$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 0.937·5-s + 6-s − 0.792·7-s − 8-s + 9-s + 0.937·10-s − 4.70·11-s − 12-s + 13-s + 0.792·14-s + 0.937·15-s + 16-s − 3.60·17-s − 18-s − 0.803·19-s − 0.937·20-s + 0.792·21-s + 4.70·22-s − 4.68·23-s + 24-s − 4.12·25-s − 26-s − 27-s − 0.792·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.419·5-s + 0.408·6-s − 0.299·7-s − 0.353·8-s + 0.333·9-s + 0.296·10-s − 1.41·11-s − 0.288·12-s + 0.277·13-s + 0.211·14-s + 0.241·15-s + 0.250·16-s − 0.874·17-s − 0.235·18-s − 0.184·19-s − 0.209·20-s + 0.172·21-s + 1.00·22-s − 0.977·23-s + 0.204·24-s − 0.824·25-s − 0.196·26-s − 0.192·27-s − 0.149·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8034} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1745776687\)
\(L(\frac12)\) \(\approx\) \(0.1745776687\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 + T \)
good5 \( 1 + 0.937T + 5T^{2} \)
7 \( 1 + 0.792T + 7T^{2} \)
11 \( 1 + 4.70T + 11T^{2} \)
17 \( 1 + 3.60T + 17T^{2} \)
19 \( 1 + 0.803T + 19T^{2} \)
23 \( 1 + 4.68T + 23T^{2} \)
29 \( 1 + 3.11T + 29T^{2} \)
31 \( 1 + 3.57T + 31T^{2} \)
37 \( 1 - 1.63T + 37T^{2} \)
41 \( 1 + 7.48T + 41T^{2} \)
43 \( 1 - 3.98T + 43T^{2} \)
47 \( 1 - 0.579T + 47T^{2} \)
53 \( 1 - 5.61T + 53T^{2} \)
59 \( 1 + 1.93T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 - 5.77T + 67T^{2} \)
71 \( 1 + 5.96T + 71T^{2} \)
73 \( 1 - 0.810T + 73T^{2} \)
79 \( 1 - 6.43T + 79T^{2} \)
83 \( 1 + 5.41T + 83T^{2} \)
89 \( 1 + 18.2T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−7.73275971890809641383897357655, −7.36679241956815382441927222551, −6.44088307217929729662718700831, −5.89814471742508835090885321558, −5.15944562936389920536260832967, −4.29218561355973869065711759080, −3.48174450359037810192729394030, −2.48638335400858269477152675958, −1.69069598273965836967808354039, −0.22739236479326189977802727441, 0.22739236479326189977802727441, 1.69069598273965836967808354039, 2.48638335400858269477152675958, 3.48174450359037810192729394030, 4.29218561355973869065711759080, 5.15944562936389920536260832967, 5.89814471742508835090885321558, 6.44088307217929729662718700831, 7.36679241956815382441927222551, 7.73275971890809641383897357655

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