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Properties

Label 2-1078-77.10-c1-0-16
Degree $2$
Conductor $1078$
Sign $0.177 + 0.984i$
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 + (−2.26 − 1.30i)3-s + (0.499 − 0.866i)4-s + (−2.26 + 1.30i)5-s − 2.61·6-s − 0.999i·8-s + (1.91 + 3.31i)9-s + (−1.30 + 2.26i)10-s + (1.89 + 2.72i)11-s + (−2.26 + 1.30i)12-s + 5.54·13-s + 6.82·15-s + (−0.5 − 0.866i)16-s + (3.31 + 1.91i)18-s + (−1.14 − 1.98i)19-s + 2.61i·20-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−1.30 − 0.754i)3-s + (0.249 − 0.433i)4-s + (−1.01 + 0.584i)5-s − 1.06·6-s − 0.353i·8-s + (0.638 + 1.10i)9-s + (−0.413 + 0.715i)10-s + (0.570 + 0.821i)11-s + (−0.653 + 0.377i)12-s + 1.53·13-s + 1.76·15-s + (−0.125 − 0.216i)16-s + (0.781 + 0.451i)18-s + (−0.263 − 0.456i)19-s + 0.584i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.177 + 0.984i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.177 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.203760775\)
\(L(\frac12)\) \(\approx\) \(1.203760775\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.89 - 2.72i)T \)
good3 \( 1 + (2.26 + 1.30i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.26 - 1.30i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 5.54T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.14 + 1.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.12 - 1.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 + (6.12 + 3.53i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + (-5.73 + 3.31i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.24 + 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.5 - 6.08i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.14 + 1.98i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.171 - 0.297i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (3.24 - 5.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.34 - 4.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.951T + 83T^{2} \)
89 \( 1 + (-10.6 + 6.15i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.09iT - 97T^{2} \)
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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

−10.06103412723706578069599750179, −8.871359785777096995277941420700, −7.57810037785787029982102365365, −7.07707005142876789720017242042, −6.19981046001389595518117648857, −5.60931326726869018319296724433, −4.29293014575923094564447128391, −3.69744801372755350386735382029, −2.09084439315093359467726047479, −0.71277036366762457584095503138, 0.994314184749903193851462234162, 3.49639979535685235129105285567, 4.01363666220255099634389740010, 4.89733827150448063725644099300, 5.74729016383718184813900384319, 6.34620998354425019958097073247, 7.38338895388735436902036409006, 8.584475842971745754335220790966, 8.942852818031692034817514400707, 10.55643400200854864642623655038

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