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Properties

Label 2-1078-77.54-c1-0-31
Degree $2$
Conductor $1078$
Sign $0.699 + 0.714i$
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 + (−3.30 + 0.275i)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 + (−0.996 + 0.0829i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.699 + 0.714i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.699 + 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.282035764\)
\(L(\frac12)\) \(\approx\) \(2.282035764\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.30 - 0.275i)T \)
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

−9.722643056064879865671485328735, −8.888935753486048035350879402264, −8.121630269558382651600832907457, −7.67722121689152340733389145386, −6.48067641404498519265979902462, −5.94155334509293992126239123819, −4.13833265242494944062806230928, −2.87095505934763552587569735444, −2.41584969384877357872164133638, −1.31960585963862133169350811951, 1.42829569671548039033924444181, 2.59685488224339480700577240803, 3.60650431768009361611345257508, 4.89843283274093002664298984406, 5.65559962856742267358968104888, 6.72548195840846309689584243681, 7.906604992387181355497208833206, 8.585154744926680331664403337133, 9.061163076654654132284712793896, 9.672271302628293972093044989451

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