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 − 3.21·5-s + 6-s − 1.40·7-s − 8-s + 9-s + 3.21·10-s − 2.53·11-s − 12-s + 13-s + 1.40·14-s + 3.21·15-s + 16-s + 7.57·17-s − 18-s + 2.48·19-s − 3.21·20-s + 1.40·21-s + 2.53·22-s + 7.10·23-s + 24-s + 5.33·25-s − 26-s − 27-s − 1.40·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.43·5-s + 0.408·6-s − 0.531·7-s − 0.353·8-s + 0.333·9-s + 1.01·10-s − 0.764·11-s − 0.288·12-s + 0.277·13-s + 0.375·14-s + 0.830·15-s + 0.250·16-s + 1.83·17-s − 0.235·18-s + 0.569·19-s − 0.718·20-s + 0.306·21-s + 0.540·22-s + 1.48·23-s + 0.204·24-s + 1.06·25-s − 0.196·26-s − 0.192·27-s − 0.265·28-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

\(L(1)\) \(\approx\) \(0.6402686369\)
\(L(\frac12)\) \(\approx\) \(0.6402686369\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.21T + 5T^{2} \)
7 \( 1 + 1.40T + 7T^{2} \)
11 \( 1 + 2.53T + 11T^{2} \)
17 \( 1 - 7.57T + 17T^{2} \)
19 \( 1 - 2.48T + 19T^{2} \)
23 \( 1 - 7.10T + 23T^{2} \)
29 \( 1 + 1.09T + 29T^{2} \)
31 \( 1 + 2.34T + 31T^{2} \)
37 \( 1 - 5.11T + 37T^{2} \)
41 \( 1 + 0.317T + 41T^{2} \)
43 \( 1 - 9.60T + 43T^{2} \)
47 \( 1 + 5.52T + 47T^{2} \)
53 \( 1 + 4.04T + 53T^{2} \)
59 \( 1 - 0.591T + 59T^{2} \)
61 \( 1 + 0.0491T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + 16.7T + 71T^{2} \)
73 \( 1 - 5.34T + 73T^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 - 3.14T + 83T^{2} \)
89 \( 1 + 4.41T + 89T^{2} \)
97 \( 1 + 9.59T + 97T^{2} \)
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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

−7.70753682739136832049616097687, −7.42498899410928935122183102910, −6.64069146383162846363881915193, −5.75980472462594208864190586046, −5.16236954197431322887815755235, −4.23775812098959791751018217350, −3.31045039073143783451033331805, −2.92613584974200004010742514198, −1.31633656835623084207108412431, −0.50590575184155791352023525741, 0.50590575184155791352023525741, 1.31633656835623084207108412431, 2.92613584974200004010742514198, 3.31045039073143783451033331805, 4.23775812098959791751018217350, 5.16236954197431322887815755235, 5.75980472462594208864190586046, 6.64069146383162846363881915193, 7.42498899410928935122183102910, 7.70753682739136832049616097687

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