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 + 1.97·5-s − 6-s + 4.65·7-s + 8-s + 9-s + 1.97·10-s + 2.35·11-s − 12-s − 13-s + 4.65·14-s − 1.97·15-s + 16-s − 1.40·17-s + 18-s − 4.22·19-s + 1.97·20-s − 4.65·21-s + 2.35·22-s − 1.68·23-s − 24-s − 1.09·25-s − 26-s − 27-s + 4.65·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.883·5-s − 0.408·6-s + 1.75·7-s + 0.353·8-s + 0.333·9-s + 0.624·10-s + 0.710·11-s − 0.288·12-s − 0.277·13-s + 1.24·14-s − 0.510·15-s + 0.250·16-s − 0.339·17-s + 0.235·18-s − 0.969·19-s + 0.441·20-s − 1.01·21-s + 0.502·22-s − 0.352·23-s − 0.204·24-s − 0.219·25-s − 0.196·26-s − 0.192·27-s + 0.879·28-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

\(L(1)\) \(\approx\) \(4.316755040\)
\(L(\frac12)\) \(\approx\) \(4.316755040\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.97T + 5T^{2} \)
7 \( 1 - 4.65T + 7T^{2} \)
11 \( 1 - 2.35T + 11T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
19 \( 1 + 4.22T + 19T^{2} \)
23 \( 1 + 1.68T + 23T^{2} \)
29 \( 1 - 0.920T + 29T^{2} \)
31 \( 1 + 4.23T + 31T^{2} \)
37 \( 1 - 1.22T + 37T^{2} \)
41 \( 1 + 8.28T + 41T^{2} \)
43 \( 1 - 9.40T + 43T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 - 4.69T + 59T^{2} \)
61 \( 1 + 1.74T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 6.96T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + 4.59T + 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 - 3.99T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−7.66558256652507730742505756097, −6.99069771083060843677648069722, −6.29446445713303262559950366216, −5.54430308366469998156183834182, −5.20859329627663376284047782558, −4.28940144700682542177144904626, −3.93649482942917825793480685545, −2.30852178544222191466563083330, −1.98852152092006543887556416834, −1.00632472357163755265413174734, 1.00632472357163755265413174734, 1.98852152092006543887556416834, 2.30852178544222191466563083330, 3.93649482942917825793480685545, 4.28940144700682542177144904626, 5.20859329627663376284047782558, 5.54430308366469998156183834182, 6.29446445713303262559950366216, 6.99069771083060843677648069722, 7.66558256652507730742505756097

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