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Properties

Label 2-1014-39.8-c1-0-44
Degree $2$
Conductor $1014$
Sign $0.235 + 0.971i$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
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  + (−0.707 + 0.707i)2-s + (1.29 − 1.15i)3-s − 1.00i·4-s + (1.82 − 1.82i)5-s + (−0.0980 + 1.72i)6-s + (2.63 − 2.63i)7-s + (0.707 + 0.707i)8-s + (0.339 − 2.98i)9-s + 2.58i·10-s + (−2.30 − 2.30i)11-s + (−1.15 − 1.29i)12-s + 3.72i·14-s + (0.253 − 4.46i)15-s − 1.00·16-s + 1.34·17-s + (1.86 + 2.34i)18-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.746 − 0.665i)3-s − 0.500i·4-s + (0.817 − 0.817i)5-s + (−0.0400 + 0.705i)6-s + (0.994 − 0.994i)7-s + (0.250 + 0.250i)8-s + (0.113 − 0.993i)9-s + 0.817i·10-s + (−0.695 − 0.695i)11-s + (−0.332 − 0.373i)12-s + 0.994i·14-s + (0.0654 − 1.15i)15-s − 0.250·16-s + 0.326·17-s + (0.440 + 0.553i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.235 + 0.971i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.235 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.969693907\)
\(L(\frac12)\) \(\approx\) \(1.969693907\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-1.29 + 1.15i)T \)
13 \( 1 \)
good5 \( 1 + (-1.82 + 1.82i)T - 5iT^{2} \)
7 \( 1 + (-2.63 + 2.63i)T - 7iT^{2} \)
11 \( 1 + (2.30 + 2.30i)T + 11iT^{2} \)
17 \( 1 - 1.34T + 17T^{2} \)
19 \( 1 + (-3.58 - 3.58i)T + 19iT^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + (0.321 + 0.321i)T + 31iT^{2} \)
37 \( 1 + (1.95 - 1.95i)T - 37iT^{2} \)
41 \( 1 + (7.89 - 7.89i)T - 41iT^{2} \)
43 \( 1 - 9.10iT - 43T^{2} \)
47 \( 1 + (-4.72 - 4.72i)T + 47iT^{2} \)
53 \( 1 - 0.216iT - 53T^{2} \)
59 \( 1 + (3.65 + 3.65i)T + 59iT^{2} \)
61 \( 1 - 6.52T + 61T^{2} \)
67 \( 1 + (2.26 + 2.26i)T + 67iT^{2} \)
71 \( 1 + (-0.108 + 0.108i)T - 71iT^{2} \)
73 \( 1 + (-3.58 + 3.58i)T - 73iT^{2} \)
79 \( 1 - 4.09T + 79T^{2} \)
83 \( 1 + (-10.5 + 10.5i)T - 83iT^{2} \)
89 \( 1 + (-8.85 - 8.85i)T + 89iT^{2} \)
97 \( 1 + (0.168 + 0.168i)T + 97iT^{2} \)
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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

−9.636647701586723003370683234343, −8.768619175035705057538370625992, −7.914098785403206859200104268621, −7.73376156734803240894095692701, −6.49669899491256434985076426655, −5.58927292743304130972721964046, −4.71137457615490794732945556942, −3.32824109869031006671416294746, −1.76146415870927979208598145183, −1.04157157915828091252079632076, 2.13305027689925202800717819247, 2.35368747344032521227167065135, 3.60146256607118053279206050774, 4.92704103708364143342571619425, 5.59583830754256115355772430691, 7.06797776220728575704326315625, 7.86040602142677235917992193425, 8.682585722709897805291322204374, 9.363041468650095381426279803420, 10.21853890239138298082680268814

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