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Properties

Label 2-1078-77.10-c1-0-29
Degree $2$
Conductor $1078$
Sign $-0.592 + 0.805i$
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 + (0.559 + 0.323i)3-s + (0.499 − 0.866i)4-s + (−2.68 + 1.54i)5-s + 0.646·6-s − 0.999i·8-s + (−1.29 − 2.23i)9-s + (−1.54 + 2.68i)10-s + (−1.52 − 2.94i)11-s + (0.559 − 0.323i)12-s + 0.646·13-s − 2·15-s + (−0.5 − 0.866i)16-s + (1.87 − 3.24i)17-s + (−2.23 − 1.29i)18-s + (0.901 + 1.56i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.323 + 0.186i)3-s + (0.249 − 0.433i)4-s + (−1.19 + 0.692i)5-s + 0.263·6-s − 0.353i·8-s + (−0.430 − 0.745i)9-s + (−0.489 + 0.847i)10-s + (−0.458 − 0.888i)11-s + (0.161 − 0.0932i)12-s + 0.179·13-s − 0.516·15-s + (−0.125 − 0.216i)16-s + (0.453 − 0.785i)17-s + (−0.527 − 0.304i)18-s + (0.206 + 0.358i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.592 + 0.805i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.592 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.250773514\)
\(L(\frac12)\) \(\approx\) \(1.250773514\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.52 + 2.94i)T \)
good3 \( 1 + (-0.559 - 0.323i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.68 - 1.54i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 0.646T + 13T^{2} \)
17 \( 1 + (-1.87 + 3.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.901 - 1.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.58iT - 29T^{2} \)
31 \( 1 + (7.48 + 4.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.79 + 3.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.93T + 41T^{2} \)
43 \( 1 - 7.16iT - 43T^{2} \)
47 \( 1 + (-8.60 + 4.96i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.79 + 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.559 + 0.323i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.969 + 1.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.79 + 6.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (8.06 - 13.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.46 + 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + (8.48 - 4.89i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.50iT - 97T^{2} \)
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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

−9.727393987058334633138952377419, −8.698071200040422634994926957602, −7.932464687094492733015089997590, −7.07202263054775400988307237177, −6.12341870599821779444023317500, −5.20582518021398750756475879569, −3.82984744883634295931882053584, −3.50111370657187049043611109626, −2.52540698437774918240218203931, −0.42170164936454081909476076403, 1.79178438191096452732640328307, 3.18707279263916645296937478979, 4.08129037289889954343897933154, 4.98145656003513615490467361316, 5.66867955746765101827398957151, 7.14850480469811180697286709217, 7.58338761385845302627372606759, 8.358429634154407820378805534131, 8.973565218650888686137447651996, 10.31876510158348294426915628997

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