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 + 1.34·5-s − 6-s − 1.25·7-s + 8-s + 9-s + 1.34·10-s + 6.18·11-s − 12-s − 13-s − 1.25·14-s − 1.34·15-s + 16-s + 0.669·17-s + 18-s + 5.57·19-s + 1.34·20-s + 1.25·21-s + 6.18·22-s − 6.51·23-s − 24-s − 3.18·25-s − 26-s − 27-s − 1.25·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.602·5-s − 0.408·6-s − 0.475·7-s + 0.353·8-s + 0.333·9-s + 0.426·10-s + 1.86·11-s − 0.288·12-s − 0.277·13-s − 0.335·14-s − 0.348·15-s + 0.250·16-s + 0.162·17-s + 0.235·18-s + 1.27·19-s + 0.301·20-s + 0.274·21-s + 1.31·22-s − 1.35·23-s − 0.204·24-s − 0.636·25-s − 0.196·26-s − 0.192·27-s − 0.237·28-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

\(L(1)\) \(\approx\) \(3.507900025\)
\(L(\frac12)\) \(\approx\) \(3.507900025\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.25T + 7T^{2} \)
11 \( 1 - 6.18T + 11T^{2} \)
17 \( 1 - 0.669T + 17T^{2} \)
19 \( 1 - 5.57T + 19T^{2} \)
23 \( 1 + 6.51T + 23T^{2} \)
29 \( 1 - 7.00T + 29T^{2} \)
31 \( 1 - 9.94T + 31T^{2} \)
37 \( 1 + 6.41T + 37T^{2} \)
41 \( 1 - 3.07T + 41T^{2} \)
43 \( 1 - 5.34T + 43T^{2} \)
47 \( 1 - 6.23T + 47T^{2} \)
53 \( 1 + 8.03T + 53T^{2} \)
59 \( 1 + 3.12T + 59T^{2} \)
61 \( 1 - 1.33T + 61T^{2} \)
67 \( 1 - 8.66T + 67T^{2} \)
71 \( 1 - 6.62T + 71T^{2} \)
73 \( 1 + 2.20T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 0.863T + 83T^{2} \)
89 \( 1 - 4.65T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−7.59984842275912173084355492910, −6.83252648908363872885731049270, −6.23792421902224650435309624008, −5.96244688712686492698980115098, −5.04256706988350522806529302423, −4.30658868665501211407281707880, −3.66352415838804105250531981085, −2.78706657577675678050241579780, −1.73758536384946751461568346802, −0.911048764324824881847917150978, 0.911048764324824881847917150978, 1.73758536384946751461568346802, 2.78706657577675678050241579780, 3.66352415838804105250531981085, 4.30658868665501211407281707880, 5.04256706988350522806529302423, 5.96244688712686492698980115098, 6.23792421902224650435309624008, 6.83252648908363872885731049270, 7.59984842275912173084355492910

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