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 + 2.41·5-s + 6-s − 0.257·7-s − 8-s + 9-s − 2.41·10-s + 1.56·11-s − 12-s + 13-s + 0.257·14-s − 2.41·15-s + 16-s + 4.26·17-s − 18-s + 1.38·19-s + 2.41·20-s + 0.257·21-s − 1.56·22-s + 8.56·23-s + 24-s + 0.851·25-s − 26-s − 27-s − 0.257·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.08·5-s + 0.408·6-s − 0.0972·7-s − 0.353·8-s + 0.333·9-s − 0.764·10-s + 0.473·11-s − 0.288·12-s + 0.277·13-s + 0.0687·14-s − 0.624·15-s + 0.250·16-s + 1.03·17-s − 0.235·18-s + 0.317·19-s + 0.540·20-s + 0.0561·21-s − 0.334·22-s + 1.78·23-s + 0.204·24-s + 0.170·25-s − 0.196·26-s − 0.192·27-s − 0.0486·28-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

\(L(1)\) \(\approx\) \(1.926417171\)
\(L(\frac12)\) \(\approx\) \(1.926417171\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 5T^{2} \)
7 \( 1 + 0.257T + 7T^{2} \)
11 \( 1 - 1.56T + 11T^{2} \)
17 \( 1 - 4.26T + 17T^{2} \)
19 \( 1 - 1.38T + 19T^{2} \)
23 \( 1 - 8.56T + 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 - 7.55T + 31T^{2} \)
37 \( 1 + 2.82T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 6.51T + 43T^{2} \)
47 \( 1 - 9.53T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + 4.00T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 9.28T + 67T^{2} \)
71 \( 1 + 0.286T + 71T^{2} \)
73 \( 1 + 15.7T + 73T^{2} \)
79 \( 1 - 9.06T + 79T^{2} \)
83 \( 1 + 6.84T + 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−7.900133359114379908315264562145, −6.95466494170447354731839515064, −6.52944036502347442566351860695, −5.88258793442047478118191418969, −5.22690695539968134372981154372, −4.46653313279949315688766266207, −3.24289333867952388407368570604, −2.59089285916287069328447111170, −1.37754656900372574842551980265, −0.918449103972021885604456540543, 0.918449103972021885604456540543, 1.37754656900372574842551980265, 2.59089285916287069328447111170, 3.24289333867952388407368570604, 4.46653313279949315688766266207, 5.22690695539968134372981154372, 5.88258793442047478118191418969, 6.52944036502347442566351860695, 6.95466494170447354731839515064, 7.900133359114379908315264562145

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