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 + 3.56·7-s − 8-s + 9-s + 2·10-s + 1.56·11-s − 12-s − 13-s − 3.56·14-s + 2·15-s + 16-s + 7.56·17-s − 18-s − 2·19-s − 2·20-s − 3.56·21-s − 1.56·22-s − 5.12·23-s + 24-s − 25-s + 26-s − 27-s + 3.56·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.894·5-s + 0.408·6-s + 1.34·7-s − 0.353·8-s + 0.333·9-s + 0.632·10-s + 0.470·11-s − 0.288·12-s − 0.277·13-s − 0.951·14-s + 0.516·15-s + 0.250·16-s + 1.83·17-s − 0.235·18-s − 0.458·19-s − 0.447·20-s − 0.777·21-s − 0.332·22-s − 1.06·23-s + 0.204·24-s − 0.200·25-s + 0.196·26-s − 0.192·27-s + 0.673·28-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 \( 1 - 3.56T + 7T^{2} \)
11 \( 1 - 1.56T + 11T^{2} \)
17 \( 1 - 7.56T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 5.12T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 7.12T + 31T^{2} \)
37 \( 1 + 0.876T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 7.12T + 43T^{2} \)
47 \( 1 - 6.24T + 47T^{2} \)
53 \( 1 - 2.43T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 2.87T + 61T^{2} \)
67 \( 1 + 0.684T + 67T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 + 8.68T + 73T^{2} \)
79 \( 1 + 17.3T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−7.44612128071544665986765105257, −7.20480031889648990049588348652, −6.01634244011360865153011104285, −5.54789648427194112280809942946, −4.60151109825343086580730462966, −4.02940192310934551562604174216, −3.10018059791519605389740493651, −1.84068298358037682276259880991, −1.18486051978851374158908330869, 0, 1.18486051978851374158908330869, 1.84068298358037682276259880991, 3.10018059791519605389740493651, 4.02940192310934551562604174216, 4.60151109825343086580730462966, 5.54789648427194112280809942946, 6.01634244011360865153011104285, 7.20480031889648990049588348652, 7.44612128071544665986765105257

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