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Properties

Label 2-1078-77.10-c1-0-36
Degree $2$
Conductor $1078$
Sign $0.672 + 0.740i$
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  + (0.866 − 0.5i)2-s + (1.60 + 0.923i)3-s + (0.499 − 0.866i)4-s + (2.14 − 1.23i)5-s + 1.84·6-s − 0.999i·8-s + (0.207 + 0.358i)9-s + (1.23 − 2.14i)10-s + (−1.30 − 3.05i)11-s + (1.60 − 0.923i)12-s − 1.96·13-s + 4.56·15-s + (−0.5 − 0.866i)16-s + (2.82 − 4.89i)17-s + (0.358 + 0.207i)18-s + (0.453 + 0.785i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.923 + 0.533i)3-s + (0.249 − 0.433i)4-s + (0.957 − 0.552i)5-s + 0.754·6-s − 0.353i·8-s + (0.0690 + 0.119i)9-s + (0.390 − 0.676i)10-s + (−0.392 − 0.919i)11-s + (0.461 − 0.266i)12-s − 0.544·13-s + 1.17·15-s + (−0.125 − 0.216i)16-s + (0.685 − 1.18i)17-s + (0.0845 + 0.0488i)18-s + (0.104 + 0.180i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.672 + 0.740i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.672 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.488408615\)
\(L(\frac12)\) \(\approx\) \(3.488408615\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 \)
11 \( 1 + (1.30 + 3.05i)T \)
good3 \( 1 + (-1.60 - 0.923i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.14 + 1.23i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 1.96T + 13T^{2} \)
17 \( 1 + (-2.82 + 4.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.453 - 0.785i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.468 - 0.811i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.50iT - 29T^{2} \)
31 \( 1 + (-4.08 - 2.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.81 - 8.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.93T + 41T^{2} \)
43 \( 1 - 6.80iT - 43T^{2} \)
47 \( 1 + (-3.46 + 2.00i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.28 - 5.68i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.836 + 0.482i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.95 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.36 - 5.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + (5.43 - 9.40i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.34 + 4.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.83T + 83T^{2} \)
89 \( 1 + (-11.0 + 6.36i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.82iT - 97T^{2} \)
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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

−9.841413990303390484807569334471, −9.067737202921311390020935817727, −8.409014934603884320930345337317, −7.28696754644607522145714821420, −6.08409781067141047951979764229, −5.30412246338973456164254250514, −4.55628723561295678879240326630, −3.18962115044809011677888863317, −2.77741788168920830149551271147, −1.26534794359214542013648990040, 2.02321574988156258118898878780, 2.46638501783409771873878584184, 3.65524029816781281649391666757, 4.86401752597702153098450589863, 5.85378356664691227720093981733, 6.60751766116841786485221373480, 7.58981607904734535160749303032, 7.993696582514115946851177994869, 9.115725147309330788527396667454, 9.990210700300596997383220198192

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