Properties

Label 2-1078-1.1-c3-0-18
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 + 2·3-s + 4·4-s − 18·5-s + 4·6-s + 8·8-s − 23·9-s − 36·10-s − 11·11-s + 8·12-s − 56·13-s − 36·15-s + 16·16-s − 36·17-s − 46·18-s + 28·19-s − 72·20-s − 22·22-s + 180·23-s + 16·24-s + 199·25-s − 112·26-s − 100·27-s − 54·29-s − 72·30-s + 334·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.384·3-s + 1/2·4-s − 1.60·5-s + 0.272·6-s + 0.353·8-s − 0.851·9-s − 1.13·10-s − 0.301·11-s + 0.192·12-s − 1.19·13-s − 0.619·15-s + 1/4·16-s − 0.513·17-s − 0.602·18-s + 0.338·19-s − 0.804·20-s − 0.213·22-s + 1.63·23-s + 0.136·24-s + 1.59·25-s − 0.844·26-s − 0.712·27-s − 0.345·29-s − 0.438·30-s + 1.93·31-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

\(L(2)\) \(\approx\) \(2.023085496\)
\(L(\frac12)\) \(\approx\) \(2.023085496\)
\(L(\frac{5}{2})\) 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 \( 1 - 2 T + p^{3} T^{2} \)
5 \( 1 + 18 T + p^{3} T^{2} \)
13 \( 1 + 56 T + p^{3} T^{2} \)
17 \( 1 + 36 T + p^{3} T^{2} \)
19 \( 1 - 28 T + p^{3} T^{2} \)
23 \( 1 - 180 T + p^{3} T^{2} \)
29 \( 1 + 54 T + p^{3} T^{2} \)
31 \( 1 - 334 T + p^{3} T^{2} \)
37 \( 1 - 386 T + p^{3} T^{2} \)
41 \( 1 - 444 T + p^{3} T^{2} \)
43 \( 1 + 316 T + p^{3} T^{2} \)
47 \( 1 - 402 T + p^{3} T^{2} \)
53 \( 1 + 486 T + p^{3} T^{2} \)
59 \( 1 - 282 T + p^{3} T^{2} \)
61 \( 1 + 380 T + p^{3} T^{2} \)
67 \( 1 - 176 T + p^{3} T^{2} \)
71 \( 1 + 324 T + p^{3} T^{2} \)
73 \( 1 + 800 T + p^{3} T^{2} \)
79 \( 1 + 1144 T + p^{3} T^{2} \)
83 \( 1 + 468 T + p^{3} T^{2} \)
89 \( 1 - 870 T + p^{3} T^{2} \)
97 \( 1 - 1330 T + p^{3} T^{2} \)
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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

−9.380249860443807390775142210348, −8.490057472116475048225750432027, −7.69025620025811719535231390289, −7.21863525393808401603061385515, −6.06046211026754263861693455945, −4.85660759887425468390901750799, −4.33687589954939945075272433443, −3.11466108981035720822412889890, −2.63783090092527752132049362909, −0.63234449474523011347087495623, 0.63234449474523011347087495623, 2.63783090092527752132049362909, 3.11466108981035720822412889890, 4.33687589954939945075272433443, 4.85660759887425468390901750799, 6.06046211026754263861693455945, 7.21863525393808401603061385515, 7.69025620025811719535231390289, 8.490057472116475048225750432027, 9.380249860443807390775142210348

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