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 − 0.343·5-s + 6-s + 2.39·7-s − 8-s + 9-s + 0.343·10-s + 1.78·11-s − 12-s + 13-s − 2.39·14-s + 0.343·15-s + 16-s + 5.22·17-s − 18-s − 3.94·19-s − 0.343·20-s − 2.39·21-s − 1.78·22-s + 2.02·23-s + 24-s − 4.88·25-s − 26-s − 27-s + 2.39·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.153·5-s + 0.408·6-s + 0.905·7-s − 0.353·8-s + 0.333·9-s + 0.108·10-s + 0.537·11-s − 0.288·12-s + 0.277·13-s − 0.640·14-s + 0.0885·15-s + 0.250·16-s + 1.26·17-s − 0.235·18-s − 0.905·19-s − 0.0767·20-s − 0.523·21-s − 0.379·22-s + 0.423·23-s + 0.204·24-s − 0.976·25-s − 0.196·26-s − 0.192·27-s + 0.452·28-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

\(L(1)\) \(\approx\) \(1.362597841\)
\(L(\frac12)\) \(\approx\) \(1.362597841\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.343T + 5T^{2} \)
7 \( 1 - 2.39T + 7T^{2} \)
11 \( 1 - 1.78T + 11T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
19 \( 1 + 3.94T + 19T^{2} \)
23 \( 1 - 2.02T + 23T^{2} \)
29 \( 1 + 7.94T + 29T^{2} \)
31 \( 1 - 2.26T + 31T^{2} \)
37 \( 1 - 8.14T + 37T^{2} \)
41 \( 1 + 0.158T + 41T^{2} \)
43 \( 1 - 2.52T + 43T^{2} \)
47 \( 1 - 7.07T + 47T^{2} \)
53 \( 1 - 6.10T + 53T^{2} \)
59 \( 1 + 8.74T + 59T^{2} \)
61 \( 1 - 3.19T + 61T^{2} \)
67 \( 1 - 7.84T + 67T^{2} \)
71 \( 1 - 7.09T + 71T^{2} \)
73 \( 1 - 17.0T + 73T^{2} \)
79 \( 1 + 3.08T + 79T^{2} \)
83 \( 1 + 9.65T + 83T^{2} \)
89 \( 1 - 6.89T + 89T^{2} \)
97 \( 1 - 7.83T + 97T^{2} \)
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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

−7.85569861008566060940843725281, −7.32117110717797596217610412364, −6.46070050858439558206955861745, −5.82403893197717985714058663478, −5.20374131983726350364225068423, −4.23002944880578177590308185026, −3.63349135513609992653832056222, −2.38139304631469384099528664002, −1.53885317331267813843548413787, −0.70820018753726197319537706233, 0.70820018753726197319537706233, 1.53885317331267813843548413787, 2.38139304631469384099528664002, 3.63349135513609992653832056222, 4.23002944880578177590308185026, 5.20374131983726350364225068423, 5.82403893197717985714058663478, 6.46070050858439558206955861745, 7.32117110717797596217610412364, 7.85569861008566060940843725281

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