Properties

Degree $2$
Conductor $8034$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 1.73·5-s − 6-s − 3.07·7-s − 8-s + 9-s + 1.73·10-s − 1.39·11-s + 12-s − 13-s + 3.07·14-s − 1.73·15-s + 16-s − 2.86·17-s − 18-s + 7.95·19-s − 1.73·20-s − 3.07·21-s + 1.39·22-s + 7.53·23-s − 24-s − 1.98·25-s + 26-s + 27-s − 3.07·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.776·5-s − 0.408·6-s − 1.16·7-s − 0.353·8-s + 0.333·9-s + 0.548·10-s − 0.419·11-s + 0.288·12-s − 0.277·13-s + 0.821·14-s − 0.448·15-s + 0.250·16-s − 0.694·17-s − 0.235·18-s + 1.82·19-s − 0.388·20-s − 0.671·21-s + 0.296·22-s + 1.57·23-s − 0.204·24-s − 0.397·25-s + 0.196·26-s + 0.192·27-s − 0.581·28-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: \(1\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T + 5T^{2} \)
7 \( 1 + 3.07T + 7T^{2} \)
11 \( 1 + 1.39T + 11T^{2} \)
17 \( 1 + 2.86T + 17T^{2} \)
19 \( 1 - 7.95T + 19T^{2} \)
23 \( 1 - 7.53T + 23T^{2} \)
29 \( 1 - 4.50T + 29T^{2} \)
31 \( 1 + 2.18T + 31T^{2} \)
37 \( 1 + 2.08T + 37T^{2} \)
41 \( 1 + 9.35T + 41T^{2} \)
43 \( 1 - 4.36T + 43T^{2} \)
47 \( 1 + 1.82T + 47T^{2} \)
53 \( 1 - 2.12T + 53T^{2} \)
59 \( 1 + 8.76T + 59T^{2} \)
61 \( 1 + 4.02T + 61T^{2} \)
67 \( 1 - 1.69T + 67T^{2} \)
71 \( 1 - 7.75T + 71T^{2} \)
73 \( 1 - 1.32T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 2.35T + 83T^{2} \)
89 \( 1 - 2.39T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−7.52348734309488637856309568949, −7.01742323992361985751323108851, −6.43940873596427807547213875674, −5.38344827435516851169284002013, −4.64794872121918258851786058292, −3.43054659827374999660483343589, −3.23205005510216568665041350903, −2.32696280735106909493050351957, −1.06282849101535722597941284503, 0, 1.06282849101535722597941284503, 2.32696280735106909493050351957, 3.23205005510216568665041350903, 3.43054659827374999660483343589, 4.64794872121918258851786058292, 5.38344827435516851169284002013, 6.43940873596427807547213875674, 7.01742323992361985751323108851, 7.52348734309488637856309568949

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