Properties

Label 2-1078-77.76-c1-0-24
Degree $2$
Conductor $1078$
Sign $0.999 - 0.0190i$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 1.08i·3-s − 4-s + 1.08i·5-s + 1.08·6-s i·8-s + 1.82·9-s − 1.08·10-s + (−1.41 − 3i)11-s + 1.08i·12-s + 2.29·13-s + 1.17·15-s + 16-s + 1.82i·18-s − 5.54·19-s − 1.08i·20-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.624i·3-s − 0.5·4-s + 0.484i·5-s + 0.441·6-s − 0.353i·8-s + 0.609·9-s − 0.342·10-s + (−0.426 − 0.904i)11-s + 0.312i·12-s + 0.636·13-s + 0.302·15-s + 0.250·16-s + 0.430i·18-s − 1.27·19-s − 0.242i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.999 - 0.0190i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.999 - 0.0190i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.587015274\)
\(L(\frac12)\) \(\approx\) \(1.587015274\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
7 \( 1 \)
11 \( 1 + (1.41 + 3i)T \)
good3 \( 1 + 1.08iT - 3T^{2} \)
5 \( 1 - 1.08iT - 5T^{2} \)
13 \( 1 - 2.29T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.54T + 19T^{2} \)
23 \( 1 - 6.24T + 23T^{2} \)
29 \( 1 + 1.75iT - 29T^{2} \)
31 \( 1 + 1.39iT - 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 4.59T + 41T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 - 4.90iT - 47T^{2} \)
53 \( 1 - 4.48T + 53T^{2} \)
59 \( 1 + 7.20iT - 59T^{2} \)
61 \( 1 + 5.54T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 - 8.48iT - 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 3.56iT - 89T^{2} \)
97 \( 1 + 8.60iT - 97T^{2} \)
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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

−9.793612545367514086572783198431, −8.779824253908352897297295442450, −8.141047055878656562820321289681, −7.28057939972844566659464079678, −6.55571217358634870123075991514, −5.95012341784612875278311110696, −4.78334061025514643307977781727, −3.72210776171302024877970478964, −2.49441296138584440191662549811, −0.884864412827685459142034235266, 1.19270590449682570131363329530, 2.48763245035909541536286516443, 3.75745542300058965709636147425, 4.57523202526214433644734184946, 5.14747272142092290461806656068, 6.49220980844134114286955641392, 7.48103294025445905329199330572, 8.577291415490005276697321231094, 9.171824366329973538055539686503, 9.966673819945883621841304462728

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