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 − 1.18·5-s − 6-s + 0.107·7-s − 8-s + 9-s + 1.18·10-s + 0.456·11-s + 12-s + 13-s − 0.107·14-s − 1.18·15-s + 16-s − 3.46·17-s − 18-s − 0.368·19-s − 1.18·20-s + 0.107·21-s − 0.456·22-s + 5.15·23-s − 24-s − 3.58·25-s − 26-s + 27-s + 0.107·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.531·5-s − 0.408·6-s + 0.0404·7-s − 0.353·8-s + 0.333·9-s + 0.375·10-s + 0.137·11-s + 0.288·12-s + 0.277·13-s − 0.0286·14-s − 0.306·15-s + 0.250·16-s − 0.841·17-s − 0.235·18-s − 0.0846·19-s − 0.265·20-s + 0.0233·21-s − 0.0973·22-s + 1.07·23-s − 0.204·24-s − 0.717·25-s − 0.196·26-s + 0.192·27-s + 0.0202·28-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 \( 1 + 1.18T + 5T^{2} \)
7 \( 1 - 0.107T + 7T^{2} \)
11 \( 1 - 0.456T + 11T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 0.368T + 19T^{2} \)
23 \( 1 - 5.15T + 23T^{2} \)
29 \( 1 + 3.44T + 29T^{2} \)
31 \( 1 - 0.0661T + 31T^{2} \)
37 \( 1 - 4.66T + 37T^{2} \)
41 \( 1 - 0.724T + 41T^{2} \)
43 \( 1 - 3.85T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 7.67T + 53T^{2} \)
59 \( 1 - 4.84T + 59T^{2} \)
61 \( 1 - 6.69T + 61T^{2} \)
67 \( 1 - 2.46T + 67T^{2} \)
71 \( 1 - 6.86T + 71T^{2} \)
73 \( 1 + 7.04T + 73T^{2} \)
79 \( 1 - 0.265T + 79T^{2} \)
83 \( 1 - 0.324T + 83T^{2} \)
89 \( 1 + 4.77T + 89T^{2} \)
97 \( 1 - 3.46T + 97T^{2} \)
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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

−7.72765948043621793360188053133, −6.88863973628612146483436493902, −6.42830784762656074955815774977, −5.42466092839219733326348445762, −4.52157709869060984406451134236, −3.78476675282600859085506166686, −3.02087730712551374560982209780, −2.16069953989833725867600471252, −1.25124801471604180707492844182, 0, 1.25124801471604180707492844182, 2.16069953989833725867600471252, 3.02087730712551374560982209780, 3.78476675282600859085506166686, 4.52157709869060984406451134236, 5.42466092839219733326348445762, 6.42830784762656074955815774977, 6.88863973628612146483436493902, 7.72765948043621793360188053133

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