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 − 0.113·5-s − 6-s + 1.75·7-s + 8-s + 9-s − 0.113·10-s + 3.65·11-s − 12-s − 13-s + 1.75·14-s + 0.113·15-s + 16-s + 6.05·17-s + 18-s − 7.50·19-s − 0.113·20-s − 1.75·21-s + 3.65·22-s + 5.43·23-s − 24-s − 4.98·25-s − 26-s − 27-s + 1.75·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0506·5-s − 0.408·6-s + 0.661·7-s + 0.353·8-s + 0.333·9-s − 0.0358·10-s + 1.10·11-s − 0.288·12-s − 0.277·13-s + 0.467·14-s + 0.0292·15-s + 0.250·16-s + 1.46·17-s + 0.235·18-s − 1.72·19-s − 0.0253·20-s − 0.381·21-s + 0.779·22-s + 1.13·23-s − 0.204·24-s − 0.997·25-s − 0.196·26-s − 0.192·27-s + 0.330·28-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

\(L(1)\) \(\approx\) \(3.391767074\)
\(L(\frac12)\) \(\approx\) \(3.391767074\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.113T + 5T^{2} \)
7 \( 1 - 1.75T + 7T^{2} \)
11 \( 1 - 3.65T + 11T^{2} \)
17 \( 1 - 6.05T + 17T^{2} \)
19 \( 1 + 7.50T + 19T^{2} \)
23 \( 1 - 5.43T + 23T^{2} \)
29 \( 1 - 9.14T + 29T^{2} \)
31 \( 1 - 4.35T + 31T^{2} \)
37 \( 1 - 0.0304T + 37T^{2} \)
41 \( 1 - 5.21T + 41T^{2} \)
43 \( 1 - 6.58T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 + 3.10T + 53T^{2} \)
59 \( 1 - 6.97T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + 5.75T + 71T^{2} \)
73 \( 1 + 6.79T + 73T^{2} \)
79 \( 1 + 5.30T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−7.76412452874756284773486798804, −6.91333369298766479805453638343, −6.33510764726185855938169610472, −5.80595867398435634329633058532, −4.85265230449767254982230121589, −4.48766927178327767997297665103, −3.69181504130747012289246272713, −2.76662305662295243696522459408, −1.72373321564844621026841628982, −0.899564883750234267537441606738, 0.899564883750234267537441606738, 1.72373321564844621026841628982, 2.76662305662295243696522459408, 3.69181504130747012289246272713, 4.48766927178327767997297665103, 4.85265230449767254982230121589, 5.80595867398435634329633058532, 6.33510764726185855938169610472, 6.91333369298766479805453638343, 7.76412452874756284773486798804

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