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 + 3.01·5-s − 6-s − 0.871·7-s + 8-s + 9-s + 3.01·10-s − 3.54·11-s − 12-s − 13-s − 0.871·14-s − 3.01·15-s + 16-s − 1.08·17-s + 18-s − 2.36·19-s + 3.01·20-s + 0.871·21-s − 3.54·22-s + 6.14·23-s − 24-s + 4.06·25-s − 26-s − 27-s − 0.871·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.34·5-s − 0.408·6-s − 0.329·7-s + 0.353·8-s + 0.333·9-s + 0.952·10-s − 1.06·11-s − 0.288·12-s − 0.277·13-s − 0.233·14-s − 0.777·15-s + 0.250·16-s − 0.263·17-s + 0.235·18-s − 0.542·19-s + 0.673·20-s + 0.190·21-s − 0.756·22-s + 1.28·23-s − 0.204·24-s + 0.813·25-s − 0.196·26-s − 0.192·27-s − 0.164·28-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

\(L(1)\) \(\approx\) \(3.109304002\)
\(L(\frac12)\) \(\approx\) \(3.109304002\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.01T + 5T^{2} \)
7 \( 1 + 0.871T + 7T^{2} \)
11 \( 1 + 3.54T + 11T^{2} \)
17 \( 1 + 1.08T + 17T^{2} \)
19 \( 1 + 2.36T + 19T^{2} \)
23 \( 1 - 6.14T + 23T^{2} \)
29 \( 1 + 2.17T + 29T^{2} \)
31 \( 1 - 0.257T + 31T^{2} \)
37 \( 1 + 2.01T + 37T^{2} \)
41 \( 1 - 8.60T + 41T^{2} \)
43 \( 1 + 1.44T + 43T^{2} \)
47 \( 1 - 8.61T + 47T^{2} \)
53 \( 1 - 8.32T + 53T^{2} \)
59 \( 1 - 6.35T + 59T^{2} \)
61 \( 1 - 15.0T + 61T^{2} \)
67 \( 1 + 4.89T + 67T^{2} \)
71 \( 1 + 6.82T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 5.43T + 79T^{2} \)
83 \( 1 + 6.99T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−7.49980954579982251316724855035, −6.97540723313876045134238447831, −6.22714526531960640468592204093, −5.69573771170553515159456417842, −5.17228558236183456586761996602, −4.54755568357754320800251110723, −3.50836489319115029417007456303, −2.51918435029489344506838898339, −2.07898219009118621726868058942, −0.791121618007017702129076677312, 0.791121618007017702129076677312, 2.07898219009118621726868058942, 2.51918435029489344506838898339, 3.50836489319115029417007456303, 4.54755568357754320800251110723, 5.17228558236183456586761996602, 5.69573771170553515159456417842, 6.22714526531960640468592204093, 6.97540723313876045134238447831, 7.49980954579982251316724855035

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