Properties

Label 2-1078-1.1-c3-0-79
Degree $2$
Conductor $1078$
Sign $1$
Analytic cond. $63.6040$
Root an. cond. $7.97521$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 10·3-s + 4·4-s + 14·5-s + 20·6-s + 8·8-s + 73·9-s + 28·10-s − 11·11-s + 40·12-s + 16·13-s + 140·15-s + 16·16-s − 108·17-s + 146·18-s − 116·19-s + 56·20-s − 22·22-s + 68·23-s + 80·24-s + 71·25-s + 32·26-s + 460·27-s + 122·29-s + 280·30-s + 262·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.92·3-s + 1/2·4-s + 1.25·5-s + 1.36·6-s + 0.353·8-s + 2.70·9-s + 0.885·10-s − 0.301·11-s + 0.962·12-s + 0.341·13-s + 2.40·15-s + 1/4·16-s − 1.54·17-s + 1.91·18-s − 1.40·19-s + 0.626·20-s − 0.213·22-s + 0.616·23-s + 0.680·24-s + 0.567·25-s + 0.241·26-s + 3.27·27-s + 0.781·29-s + 1.70·30-s + 1.51·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(63.6040\)
Root analytic conductor: \(7.97521\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1078} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(8.681955880\)
\(L(\frac12)\) \(\approx\) \(8.681955880\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
7 \( 1 \)
11 \( 1 + p T \)
good3 \( 1 - 10 T + p^{3} T^{2} \)
5 \( 1 - 14 T + p^{3} T^{2} \)
13 \( 1 - 16 T + p^{3} T^{2} \)
17 \( 1 + 108 T + p^{3} T^{2} \)
19 \( 1 + 116 T + p^{3} T^{2} \)
23 \( 1 - 68 T + p^{3} T^{2} \)
29 \( 1 - 122 T + p^{3} T^{2} \)
31 \( 1 - 262 T + p^{3} T^{2} \)
37 \( 1 - 130 T + p^{3} T^{2} \)
41 \( 1 + 204 T + p^{3} T^{2} \)
43 \( 1 + 396 T + p^{3} T^{2} \)
47 \( 1 + 166 T + p^{3} T^{2} \)
53 \( 1 - 442 T + p^{3} T^{2} \)
59 \( 1 + 702 T + p^{3} T^{2} \)
61 \( 1 + 196 T + p^{3} T^{2} \)
67 \( 1 + 416 T + p^{3} T^{2} \)
71 \( 1 - 492 T + p^{3} T^{2} \)
73 \( 1 + 408 T + p^{3} T^{2} \)
79 \( 1 - 600 T + p^{3} T^{2} \)
83 \( 1 - 1212 T + p^{3} T^{2} \)
89 \( 1 + 1146 T + p^{3} T^{2} \)
97 \( 1 - 482 T + p^{3} T^{2} \)
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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

−9.395043648689765753297873444237, −8.639708608049338677771771810822, −8.084772878935851281092936147701, −6.77263049228841520704700875297, −6.39102275319412242077308119572, −4.86898295824076697637476715246, −4.17087860958537489820041766948, −2.95945468919319756239614729241, −2.34380061902347123326311305041, −1.56626390947823695689903009149, 1.56626390947823695689903009149, 2.34380061902347123326311305041, 2.95945468919319756239614729241, 4.17087860958537489820041766948, 4.86898295824076697637476715246, 6.39102275319412242077308119572, 6.77263049228841520704700875297, 8.084772878935851281092936147701, 8.639708608049338677771771810822, 9.395043648689765753297873444237

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