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 + 2.56·5-s + 6-s + 3.96·7-s − 8-s + 9-s − 2.56·10-s − 3.64·11-s − 12-s − 13-s − 3.96·14-s − 2.56·15-s + 16-s + 6.34·17-s − 18-s + 6.47·19-s + 2.56·20-s − 3.96·21-s + 3.64·22-s + 5.03·23-s + 24-s + 1.60·25-s + 26-s − 27-s + 3.96·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.14·5-s + 0.408·6-s + 1.49·7-s − 0.353·8-s + 0.333·9-s − 0.812·10-s − 1.10·11-s − 0.288·12-s − 0.277·13-s − 1.05·14-s − 0.663·15-s + 0.250·16-s + 1.53·17-s − 0.235·18-s + 1.48·19-s + 0.574·20-s − 0.864·21-s + 0.778·22-s + 1.04·23-s + 0.204·24-s + 0.320·25-s + 0.196·26-s − 0.192·27-s + 0.748·28-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

\(L(1)\) \(\approx\) \(2.139401322\)
\(L(\frac12)\) \(\approx\) \(2.139401322\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 - 3.96T + 7T^{2} \)
11 \( 1 + 3.64T + 11T^{2} \)
17 \( 1 - 6.34T + 17T^{2} \)
19 \( 1 - 6.47T + 19T^{2} \)
23 \( 1 - 5.03T + 23T^{2} \)
29 \( 1 - 4.90T + 29T^{2} \)
31 \( 1 - 8.81T + 31T^{2} \)
37 \( 1 + 7.78T + 37T^{2} \)
41 \( 1 - 2.82T + 41T^{2} \)
43 \( 1 - 8.04T + 43T^{2} \)
47 \( 1 + 5.14T + 47T^{2} \)
53 \( 1 - 0.106T + 53T^{2} \)
59 \( 1 + 6.22T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + 2.03T + 67T^{2} \)
71 \( 1 - 0.802T + 71T^{2} \)
73 \( 1 - 8.44T + 73T^{2} \)
79 \( 1 - 0.928T + 79T^{2} \)
83 \( 1 + 6.12T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 - 9.59T + 97T^{2} \)
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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

−7.69821073350271609287203610106, −7.46371927831500523844891278164, −6.39982043151793176824265656020, −5.66443664231051451794299637964, −5.13365155288714455941813030703, −4.75897381671482587437038701851, −3.19709326198635350182080402740, −2.47279350519668054954130853325, −1.46444779705088097774795430065, −0.945936293955441338155437069543, 0.945936293955441338155437069543, 1.46444779705088097774795430065, 2.47279350519668054954130853325, 3.19709326198635350182080402740, 4.75897381671482587437038701851, 5.13365155288714455941813030703, 5.66443664231051451794299637964, 6.39982043151793176824265656020, 7.46371927831500523844891278164, 7.69821073350271609287203610106

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