Properties

Degree $2$
Conductor $8034$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 0.480·5-s − 6-s + 0.765·7-s − 8-s + 9-s − 0.480·10-s − 1.12·11-s + 12-s − 13-s − 0.765·14-s + 0.480·15-s + 16-s + 0.131·17-s − 18-s + 7.23·19-s + 0.480·20-s + 0.765·21-s + 1.12·22-s − 6.47·23-s − 24-s − 4.76·25-s + 26-s + 27-s + 0.765·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.214·5-s − 0.408·6-s + 0.289·7-s − 0.353·8-s + 0.333·9-s − 0.152·10-s − 0.338·11-s + 0.288·12-s − 0.277·13-s − 0.204·14-s + 0.124·15-s + 0.250·16-s + 0.0318·17-s − 0.235·18-s + 1.65·19-s + 0.107·20-s + 0.167·21-s + 0.239·22-s − 1.35·23-s − 0.204·24-s − 0.953·25-s + 0.196·26-s + 0.192·27-s + 0.144·28-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: \(1\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.480T + 5T^{2} \)
7 \( 1 - 0.765T + 7T^{2} \)
11 \( 1 + 1.12T + 11T^{2} \)
17 \( 1 - 0.131T + 17T^{2} \)
19 \( 1 - 7.23T + 19T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 + 5.92T + 29T^{2} \)
31 \( 1 - 0.823T + 31T^{2} \)
37 \( 1 + 5.74T + 37T^{2} \)
41 \( 1 - 9.71T + 41T^{2} \)
43 \( 1 + 1.79T + 43T^{2} \)
47 \( 1 + 1.21T + 47T^{2} \)
53 \( 1 + 6.16T + 53T^{2} \)
59 \( 1 + 7.84T + 59T^{2} \)
61 \( 1 - 8.36T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 16.6T + 71T^{2} \)
73 \( 1 + 2.44T + 73T^{2} \)
79 \( 1 - 7.55T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
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

−7.74969373064232638267315070516, −7.11713581824013246217840309899, −6.09353123158311738832950874937, −5.55789608892876491965656452513, −4.65645705847844051006551848208, −3.70649781194196757051952799354, −2.97815033207879842142089337210, −2.07270200427862947148767780617, −1.40145604728960189542194688654, 0, 1.40145604728960189542194688654, 2.07270200427862947148767780617, 2.97815033207879842142089337210, 3.70649781194196757051952799354, 4.65645705847844051006551848208, 5.55789608892876491965656452513, 6.09353123158311738832950874937, 7.11713581824013246217840309899, 7.74969373064232638267315070516

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