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 + 4.38·5-s + 6-s + 3.17·7-s − 8-s + 9-s − 4.38·10-s − 0.317·11-s − 12-s + 13-s − 3.17·14-s − 4.38·15-s + 16-s − 0.0669·17-s − 18-s − 0.603·19-s + 4.38·20-s − 3.17·21-s + 0.317·22-s − 1.74·23-s + 24-s + 14.2·25-s − 26-s − 27-s + 3.17·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.96·5-s + 0.408·6-s + 1.20·7-s − 0.353·8-s + 0.333·9-s − 1.38·10-s − 0.0956·11-s − 0.288·12-s + 0.277·13-s − 0.849·14-s − 1.13·15-s + 0.250·16-s − 0.0162·17-s − 0.235·18-s − 0.138·19-s + 0.981·20-s − 0.693·21-s + 0.0676·22-s − 0.364·23-s + 0.204·24-s + 2.85·25-s − 0.196·26-s − 0.192·27-s + 0.600·28-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

\(L(1)\) \(\approx\) \(2.275023123\)
\(L(\frac12)\) \(\approx\) \(2.275023123\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.38T + 5T^{2} \)
7 \( 1 - 3.17T + 7T^{2} \)
11 \( 1 + 0.317T + 11T^{2} \)
17 \( 1 + 0.0669T + 17T^{2} \)
19 \( 1 + 0.603T + 19T^{2} \)
23 \( 1 + 1.74T + 23T^{2} \)
29 \( 1 + 3.43T + 29T^{2} \)
31 \( 1 - 8.86T + 31T^{2} \)
37 \( 1 + 2.15T + 37T^{2} \)
41 \( 1 + 5.16T + 41T^{2} \)
43 \( 1 + 7.69T + 43T^{2} \)
47 \( 1 + 9.65T + 47T^{2} \)
53 \( 1 - 1.21T + 53T^{2} \)
59 \( 1 - 3.33T + 59T^{2} \)
61 \( 1 + 1.56T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 - 7.48T + 71T^{2} \)
73 \( 1 - 1.78T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 + 5.61T + 89T^{2} \)
97 \( 1 + 14.4T + 97T^{2} \)
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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

−8.103360409712559009380912526955, −6.91266366561723905163524700422, −6.50807472661907161562427977249, −5.80565041862816767129765311428, −5.16164106024411762269158530743, −4.68398445377258213210308767654, −3.29428031189360490232618927302, −2.13542627773650715616444668252, −1.77789718395072637362065327231, −0.903478823951766187924656229891, 0.903478823951766187924656229891, 1.77789718395072637362065327231, 2.13542627773650715616444668252, 3.29428031189360490232618927302, 4.68398445377258213210308767654, 5.16164106024411762269158530743, 5.80565041862816767129765311428, 6.50807472661907161562427977249, 6.91266366561723905163524700422, 8.103360409712559009380912526955

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