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 − 2·5-s + 6-s − 0.561·7-s − 8-s + 9-s + 2·10-s − 2.56·11-s − 12-s − 13-s + 0.561·14-s + 2·15-s + 16-s + 3.43·17-s − 18-s − 2·19-s − 2·20-s + 0.561·21-s + 2.56·22-s + 3.12·23-s + 24-s − 25-s + 26-s − 27-s − 0.561·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.894·5-s + 0.408·6-s − 0.212·7-s − 0.353·8-s + 0.333·9-s + 0.632·10-s − 0.772·11-s − 0.288·12-s − 0.277·13-s + 0.150·14-s + 0.516·15-s + 0.250·16-s + 0.833·17-s − 0.235·18-s − 0.458·19-s − 0.447·20-s + 0.122·21-s + 0.546·22-s + 0.651·23-s + 0.204·24-s − 0.200·25-s + 0.196·26-s − 0.192·27-s − 0.106·28-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 \( 1 + 2T + 5T^{2} \)
7 \( 1 + 0.561T + 7T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
17 \( 1 - 3.43T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 3.12T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 1.12T + 31T^{2} \)
37 \( 1 + 9.12T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 1.12T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 6.56T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 - 2.24T + 71T^{2} \)
73 \( 1 - 3.68T + 73T^{2} \)
79 \( 1 - 7.36T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−7.61504821740299154111283982069, −6.85137825167425043529220644590, −6.33920176261175380405005760155, −5.31751596245021399152497590244, −4.85432041254355892840383758923, −3.78794597914259642695522924027, −3.10912921376212607483798009680, −2.10981902749199411924745735354, −0.909462209393519328707906409985, 0, 0.909462209393519328707906409985, 2.10981902749199411924745735354, 3.10912921376212607483798009680, 3.78794597914259642695522924027, 4.85432041254355892840383758923, 5.31751596245021399152497590244, 6.33920176261175380405005760155, 6.85137825167425043529220644590, 7.61504821740299154111283982069

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