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 − 3.65·5-s + 6-s − 2.98·7-s − 8-s + 9-s + 3.65·10-s + 3.60·11-s − 12-s + 13-s + 2.98·14-s + 3.65·15-s + 16-s − 3.22·17-s − 18-s + 3.44·19-s − 3.65·20-s + 2.98·21-s − 3.60·22-s + 3.62·23-s + 24-s + 8.35·25-s − 26-s − 27-s − 2.98·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.63·5-s + 0.408·6-s − 1.12·7-s − 0.353·8-s + 0.333·9-s + 1.15·10-s + 1.08·11-s − 0.288·12-s + 0.277·13-s + 0.797·14-s + 0.943·15-s + 0.250·16-s − 0.781·17-s − 0.235·18-s + 0.791·19-s − 0.817·20-s + 0.651·21-s − 0.769·22-s + 0.755·23-s + 0.204·24-s + 1.67·25-s − 0.196·26-s − 0.192·27-s − 0.564·28-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

\(L(1)\) \(\approx\) \(0.6018093083\)
\(L(\frac12)\) \(\approx\) \(0.6018093083\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.65T + 5T^{2} \)
7 \( 1 + 2.98T + 7T^{2} \)
11 \( 1 - 3.60T + 11T^{2} \)
17 \( 1 + 3.22T + 17T^{2} \)
19 \( 1 - 3.44T + 19T^{2} \)
23 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 1.67T + 37T^{2} \)
41 \( 1 + 8.55T + 41T^{2} \)
43 \( 1 - 5.42T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 + 1.05T + 53T^{2} \)
59 \( 1 + 5.39T + 59T^{2} \)
61 \( 1 + 6.09T + 61T^{2} \)
67 \( 1 - 0.180T + 67T^{2} \)
71 \( 1 + 2.75T + 71T^{2} \)
73 \( 1 + 6.96T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 - 4.83T + 97T^{2} \)
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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

−7.81718179373297955044422998894, −7.08638518783397380194194369710, −6.52989529474190563137172219710, −6.22221730289671379057315043878, −4.80992930383466473792396908168, −4.33864933353556614375512149727, −3.33843465109529537742336796587, −2.95145343245712398273314888586, −1.27939950253271391610590256979, −0.49925470795214414458870578708, 0.49925470795214414458870578708, 1.27939950253271391610590256979, 2.95145343245712398273314888586, 3.33843465109529537742336796587, 4.33864933353556614375512149727, 4.80992930383466473792396908168, 6.22221730289671379057315043878, 6.52989529474190563137172219710, 7.08638518783397380194194369710, 7.81718179373297955044422998894

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