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 − 2.05·5-s + 6-s + 5.03·7-s − 8-s + 9-s + 2.05·10-s + 3.25·11-s − 12-s + 13-s − 5.03·14-s + 2.05·15-s + 16-s + 0.479·17-s − 18-s + 5.58·19-s − 2.05·20-s − 5.03·21-s − 3.25·22-s − 6.13·23-s + 24-s − 0.793·25-s − 26-s − 27-s + 5.03·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.917·5-s + 0.408·6-s + 1.90·7-s − 0.353·8-s + 0.333·9-s + 0.648·10-s + 0.981·11-s − 0.288·12-s + 0.277·13-s − 1.34·14-s + 0.529·15-s + 0.250·16-s + 0.116·17-s − 0.235·18-s + 1.28·19-s − 0.458·20-s − 1.09·21-s − 0.694·22-s − 1.27·23-s + 0.204·24-s − 0.158·25-s − 0.196·26-s − 0.192·27-s + 0.951·28-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

\(L(1)\) \(\approx\) \(1.545843010\)
\(L(\frac12)\) \(\approx\) \(1.545843010\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.05T + 5T^{2} \)
7 \( 1 - 5.03T + 7T^{2} \)
11 \( 1 - 3.25T + 11T^{2} \)
17 \( 1 - 0.479T + 17T^{2} \)
19 \( 1 - 5.58T + 19T^{2} \)
23 \( 1 + 6.13T + 23T^{2} \)
29 \( 1 - 3.21T + 29T^{2} \)
31 \( 1 - 8.30T + 31T^{2} \)
37 \( 1 + 3.88T + 37T^{2} \)
41 \( 1 - 5.27T + 41T^{2} \)
43 \( 1 - 3.12T + 43T^{2} \)
47 \( 1 - 3.55T + 47T^{2} \)
53 \( 1 + 6.76T + 53T^{2} \)
59 \( 1 - 13.1T + 59T^{2} \)
61 \( 1 - 6.84T + 61T^{2} \)
67 \( 1 + 3.51T + 67T^{2} \)
71 \( 1 - 0.616T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 - 2.05T + 79T^{2} \)
83 \( 1 + 2.18T + 83T^{2} \)
89 \( 1 + 3.35T + 89T^{2} \)
97 \( 1 + 1.04T + 97T^{2} \)
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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

−7.988893472802547673960217244823, −7.34806000649957420351728852068, −6.60345633296678568297676456750, −5.76035195685537948002892881790, −5.04761735888372859756699481659, −4.25483575044017747101110045088, −3.73776453516265249758211201268, −2.40119216391729966816659620026, −1.39999677892357748464225810833, −0.811643889722881371981280178321, 0.811643889722881371981280178321, 1.39999677892357748464225810833, 2.40119216391729966816659620026, 3.73776453516265249758211201268, 4.25483575044017747101110045088, 5.04761735888372859756699481659, 5.76035195685537948002892881790, 6.60345633296678568297676456750, 7.34806000649957420351728852068, 7.988893472802547673960217244823

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