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 + 0.522·5-s + 6-s − 2.74·7-s − 8-s + 9-s − 0.522·10-s − 0.360·11-s − 12-s − 13-s + 2.74·14-s − 0.522·15-s + 16-s + 1.93·17-s − 18-s − 0.748·19-s + 0.522·20-s + 2.74·21-s + 0.360·22-s − 1.23·23-s + 24-s − 4.72·25-s + 26-s − 27-s − 2.74·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.233·5-s + 0.408·6-s − 1.03·7-s − 0.353·8-s + 0.333·9-s − 0.165·10-s − 0.108·11-s − 0.288·12-s − 0.277·13-s + 0.734·14-s − 0.134·15-s + 0.250·16-s + 0.468·17-s − 0.235·18-s − 0.171·19-s + 0.116·20-s + 0.599·21-s + 0.0768·22-s − 0.257·23-s + 0.204·24-s − 0.945·25-s + 0.196·26-s − 0.192·27-s − 0.519·28-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

\(L(1)\) \(\approx\) \(0.6094699078\)
\(L(\frac12)\) \(\approx\) \(0.6094699078\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.522T + 5T^{2} \)
7 \( 1 + 2.74T + 7T^{2} \)
11 \( 1 + 0.360T + 11T^{2} \)
17 \( 1 - 1.93T + 17T^{2} \)
19 \( 1 + 0.748T + 19T^{2} \)
23 \( 1 + 1.23T + 23T^{2} \)
29 \( 1 + 4.90T + 29T^{2} \)
31 \( 1 - 1.19T + 31T^{2} \)
37 \( 1 + 2.99T + 37T^{2} \)
41 \( 1 - 5.55T + 41T^{2} \)
43 \( 1 - 5.82T + 43T^{2} \)
47 \( 1 + 0.902T + 47T^{2} \)
53 \( 1 + 9.38T + 53T^{2} \)
59 \( 1 - 6.21T + 59T^{2} \)
61 \( 1 - 6.71T + 61T^{2} \)
67 \( 1 - 2.28T + 67T^{2} \)
71 \( 1 + 1.41T + 71T^{2} \)
73 \( 1 + 15.8T + 73T^{2} \)
79 \( 1 + 7.70T + 79T^{2} \)
83 \( 1 + 2.77T + 83T^{2} \)
89 \( 1 - 1.03T + 89T^{2} \)
97 \( 1 - 7.76T + 97T^{2} \)
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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

−7.65845301227760232724684860860, −7.26243189796902525724442816632, −6.37617229375705504252745324056, −5.95884899455880869809272441412, −5.28575572002975074706181465021, −4.22602534258585031906609322297, −3.43105457867528214748456355960, −2.54415939567930207918179983038, −1.62110962744451062155198913911, −0.43870123530732401506963949410, 0.43870123530732401506963949410, 1.62110962744451062155198913911, 2.54415939567930207918179983038, 3.43105457867528214748456355960, 4.22602534258585031906609322297, 5.28575572002975074706181465021, 5.95884899455880869809272441412, 6.37617229375705504252745324056, 7.26243189796902525724442816632, 7.65845301227760232724684860860

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