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 + 1.05·5-s + 6-s − 0.189·7-s − 8-s + 9-s − 1.05·10-s − 6.30·11-s − 12-s − 13-s + 0.189·14-s − 1.05·15-s + 16-s − 4.95·17-s − 18-s − 6.89·19-s + 1.05·20-s + 0.189·21-s + 6.30·22-s + 8.44·23-s + 24-s − 3.89·25-s + 26-s − 27-s − 0.189·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.470·5-s + 0.408·6-s − 0.0715·7-s − 0.353·8-s + 0.333·9-s − 0.333·10-s − 1.89·11-s − 0.288·12-s − 0.277·13-s + 0.0506·14-s − 0.271·15-s + 0.250·16-s − 1.20·17-s − 0.235·18-s − 1.58·19-s + 0.235·20-s + 0.0413·21-s + 1.34·22-s + 1.76·23-s + 0.204·24-s − 0.778·25-s + 0.196·26-s − 0.192·27-s − 0.0357·28-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

\(L(1)\) \(\approx\) \(0.3714993013\)
\(L(\frac12)\) \(\approx\) \(0.3714993013\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.05T + 5T^{2} \)
7 \( 1 + 0.189T + 7T^{2} \)
11 \( 1 + 6.30T + 11T^{2} \)
17 \( 1 + 4.95T + 17T^{2} \)
19 \( 1 + 6.89T + 19T^{2} \)
23 \( 1 - 8.44T + 23T^{2} \)
29 \( 1 + 2.48T + 29T^{2} \)
31 \( 1 + 6.01T + 31T^{2} \)
37 \( 1 - 0.704T + 37T^{2} \)
41 \( 1 + 0.424T + 41T^{2} \)
43 \( 1 + 5.29T + 43T^{2} \)
47 \( 1 - 4.70T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 - 1.74T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 - 8.73T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 7.04T + 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + 2.13T + 97T^{2} \)
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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

−7.954082692107398945045946624830, −7.05543776884518291026456913572, −6.60289677126376471735366429080, −5.79272514688115744234011067810, −5.12781842934853044439343886911, −4.54128137394680303489103706360, −3.31045084309185830471729142554, −2.36158893366604839381336052027, −1.84459608298948812983489107180, −0.32390962819546747079550265035, 0.32390962819546747079550265035, 1.84459608298948812983489107180, 2.36158893366604839381336052027, 3.31045084309185830471729142554, 4.54128137394680303489103706360, 5.12781842934853044439343886911, 5.79272514688115744234011067810, 6.60289677126376471735366429080, 7.05543776884518291026456913572, 7.954082692107398945045946624830

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