Properties

Label 2-8034-1.1-c1-0-2
Degree $2$
Conductor $8034$
Sign $1$
Analytic cond. $64.1518$
Root an. cond. $8.00948$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 1.34·5-s + 6-s − 3.24·7-s − 8-s + 9-s − 1.34·10-s − 3.77·11-s − 12-s + 13-s + 3.24·14-s − 1.34·15-s + 16-s − 7.24·17-s − 18-s + 0.661·19-s + 1.34·20-s + 3.24·21-s + 3.77·22-s − 3.88·23-s + 24-s − 3.18·25-s − 26-s − 27-s − 3.24·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.601·5-s + 0.408·6-s − 1.22·7-s − 0.353·8-s + 0.333·9-s − 0.425·10-s − 1.13·11-s − 0.288·12-s + 0.277·13-s + 0.867·14-s − 0.347·15-s + 0.250·16-s − 1.75·17-s − 0.235·18-s + 0.151·19-s + 0.300·20-s + 0.708·21-s + 0.804·22-s − 0.810·23-s + 0.204·24-s − 0.637·25-s − 0.196·26-s − 0.192·27-s − 0.613·28-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: $1$
Analytic conductor: \(64.1518\)
Root analytic conductor: \(8.00948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3102099440\)
\(L(\frac12)\) \(\approx\) \(0.3102099440\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.34T + 5T^{2} \)
7 \( 1 + 3.24T + 7T^{2} \)
11 \( 1 + 3.77T + 11T^{2} \)
17 \( 1 + 7.24T + 17T^{2} \)
19 \( 1 - 0.661T + 19T^{2} \)
23 \( 1 + 3.88T + 23T^{2} \)
29 \( 1 + 2.76T + 29T^{2} \)
31 \( 1 - 7.79T + 31T^{2} \)
37 \( 1 + 5.09T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 - 0.261T + 43T^{2} \)
47 \( 1 - 0.430T + 47T^{2} \)
53 \( 1 + 6.98T + 53T^{2} \)
59 \( 1 - 5.79T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 - 1.48T + 71T^{2} \)
73 \( 1 + 7.24T + 73T^{2} \)
79 \( 1 - 9.27T + 79T^{2} \)
83 \( 1 - 3.12T + 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + 4.13T + 97T^{2} \)
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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

−7.923375835640141658369531847093, −6.88800478824989784321769940072, −6.59224711959938823030230250600, −5.91321135100916864873610351141, −5.26949977481949733748276227255, −4.32723625323309499244398871124, −3.34134512121663744495903923486, −2.47771682343444644848139151165, −1.75847756131711736160238542532, −0.29849432704311060972946178695, 0.29849432704311060972946178695, 1.75847756131711736160238542532, 2.47771682343444644848139151165, 3.34134512121663744495903923486, 4.32723625323309499244398871124, 5.26949977481949733748276227255, 5.91321135100916864873610351141, 6.59224711959938823030230250600, 6.88800478824989784321769940072, 7.923375835640141658369531847093

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