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 + 2.28·5-s + 6-s − 1.19·7-s − 8-s + 9-s − 2.28·10-s + 4.27·11-s − 12-s + 13-s + 1.19·14-s − 2.28·15-s + 16-s − 3.59·17-s − 18-s − 1.35·19-s + 2.28·20-s + 1.19·21-s − 4.27·22-s − 6.42·23-s + 24-s + 0.241·25-s − 26-s − 27-s − 1.19·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.02·5-s + 0.408·6-s − 0.451·7-s − 0.353·8-s + 0.333·9-s − 0.723·10-s + 1.29·11-s − 0.288·12-s + 0.277·13-s + 0.319·14-s − 0.591·15-s + 0.250·16-s − 0.872·17-s − 0.235·18-s − 0.311·19-s + 0.511·20-s + 0.260·21-s − 0.912·22-s − 1.33·23-s + 0.204·24-s + 0.0483·25-s − 0.196·26-s − 0.192·27-s − 0.225·28-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

\(L(1)\) \(\approx\) \(1.228739397\)
\(L(\frac12)\) \(\approx\) \(1.228739397\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.28T + 5T^{2} \)
7 \( 1 + 1.19T + 7T^{2} \)
11 \( 1 - 4.27T + 11T^{2} \)
17 \( 1 + 3.59T + 17T^{2} \)
19 \( 1 + 1.35T + 19T^{2} \)
23 \( 1 + 6.42T + 23T^{2} \)
29 \( 1 + 6.53T + 29T^{2} \)
31 \( 1 + 1.25T + 31T^{2} \)
37 \( 1 + 6.96T + 37T^{2} \)
41 \( 1 - 7.36T + 41T^{2} \)
43 \( 1 - 1.70T + 43T^{2} \)
47 \( 1 - 1.19T + 47T^{2} \)
53 \( 1 - 2.84T + 53T^{2} \)
59 \( 1 - 14.3T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 1.20T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 + 2.00T + 79T^{2} \)
83 \( 1 - 4.49T + 83T^{2} \)
89 \( 1 - 5.55T + 89T^{2} \)
97 \( 1 - 7.35T + 97T^{2} \)
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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

−7.84509940883857872683386228669, −6.89515879861318731309937648239, −6.52049655421087663121516140308, −5.92987964955260696976905348554, −5.35174475465480757039955474948, −4.15540767621612142369487368984, −3.60779150013175953675925748948, −2.19814247506004274765773294742, −1.80055470522176618120902852368, −0.62445002108060695948292282719, 0.62445002108060695948292282719, 1.80055470522176618120902852368, 2.19814247506004274765773294742, 3.60779150013175953675925748948, 4.15540767621612142369487368984, 5.35174475465480757039955474948, 5.92987964955260696976905348554, 6.52049655421087663121516140308, 6.89515879861318731309937648239, 7.84509940883857872683386228669

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