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Properties

Label 2-1078-77.54-c1-0-13
Degree $2$
Conductor $1078$
Sign $0.485 - 0.874i$
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 + (3.31 + 0.155i)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 + (0.998 + 0.0469i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.485 - 0.874i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.485 - 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.321767465\)
\(L(\frac12)\) \(\approx\) \(1.321767465\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 - 0.155i)T \)
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

−10.12604937329807800476262960767, −9.433909320779816183620759428291, −8.497437282165626109929905484568, −7.64545643969916088062228776097, −6.45115598808584269252773623312, −6.05581986346336746645079654856, −4.90539842386372438391743144781, −3.57505722505425420049408456166, −2.46458192660685450404396363210, −1.47914465292805377906716923044, 0.812757489884072533201080216345, 1.83055492078967575354208415731, 3.34905375384890983202990470345, 4.86179599797760014584309931635, 5.67341475168383594282259486386, 6.39838471604359431304709479187, 7.00818059381023216413864738366, 8.434737247418659540386943707313, 8.852663542809203290390272245080, 9.736719856506523670466216745978

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