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 − 2.43·5-s − 6-s − 2.53·7-s + 8-s + 9-s − 2.43·10-s + 5.77·11-s − 12-s − 13-s − 2.53·14-s + 2.43·15-s + 16-s − 4.04·17-s + 18-s − 2.00·19-s − 2.43·20-s + 2.53·21-s + 5.77·22-s + 8.05·23-s − 24-s + 0.921·25-s − 26-s − 27-s − 2.53·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.08·5-s − 0.408·6-s − 0.959·7-s + 0.353·8-s + 0.333·9-s − 0.769·10-s + 1.74·11-s − 0.288·12-s − 0.277·13-s − 0.678·14-s + 0.628·15-s + 0.250·16-s − 0.982·17-s + 0.235·18-s − 0.460·19-s − 0.544·20-s + 0.553·21-s + 1.23·22-s + 1.67·23-s − 0.204·24-s + 0.184·25-s − 0.196·26-s − 0.192·27-s − 0.479·28-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

\(L(1)\) \(\approx\) \(1.604421953\)
\(L(\frac12)\) \(\approx\) \(1.604421953\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.43T + 5T^{2} \)
7 \( 1 + 2.53T + 7T^{2} \)
11 \( 1 - 5.77T + 11T^{2} \)
17 \( 1 + 4.04T + 17T^{2} \)
19 \( 1 + 2.00T + 19T^{2} \)
23 \( 1 - 8.05T + 23T^{2} \)
29 \( 1 - 0.475T + 29T^{2} \)
31 \( 1 + 6.32T + 31T^{2} \)
37 \( 1 + 2.13T + 37T^{2} \)
41 \( 1 - 1.29T + 41T^{2} \)
43 \( 1 + 1.58T + 43T^{2} \)
47 \( 1 - 8.72T + 47T^{2} \)
53 \( 1 - 0.740T + 53T^{2} \)
59 \( 1 + 2.73T + 59T^{2} \)
61 \( 1 + 6.04T + 61T^{2} \)
67 \( 1 + 3.11T + 67T^{2} \)
71 \( 1 - 8.83T + 71T^{2} \)
73 \( 1 + 8.59T + 73T^{2} \)
79 \( 1 + 2.59T + 79T^{2} \)
83 \( 1 + 3.99T + 83T^{2} \)
89 \( 1 + 9.66T + 89T^{2} \)
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

−7.35232243167086480099629077215, −7.08093671994428301431169734294, −6.44312051359844401208040604633, −5.87628990371147403451861218634, −4.84031544830269624827570567521, −4.24724410744567355429805488546, −3.68669993049497442217026415148, −3.00735406895563109463887377403, −1.76967532587472759928280973581, −0.57646646621508972410433127512, 0.57646646621508972410433127512, 1.76967532587472759928280973581, 3.00735406895563109463887377403, 3.68669993049497442217026415148, 4.24724410744567355429805488546, 4.84031544830269624827570567521, 5.87628990371147403451861218634, 6.44312051359844401208040604633, 7.08093671994428301431169734294, 7.35232243167086480099629077215

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