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Properties

Label 2-1027-1027.315-c0-0-2
Degree $2$
Conductor $1027$
Sign $0.252 - 0.967i$
Analytic cond. $0.512539$
Root an. cond. $0.715918$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s − 5-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 13-s + (0.5 + 0.866i)16-s − 0.999·18-s + (0.5 − 0.866i)19-s + (−0.499 + 0.866i)22-s + (0.5 + 0.866i)23-s + (0.5 + 0.866i)26-s − 31-s + 0.999·38-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s − 5-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 13-s + (0.5 + 0.866i)16-s − 0.999·18-s + (0.5 − 0.866i)19-s + (−0.499 + 0.866i)22-s + (0.5 + 0.866i)23-s + (0.5 + 0.866i)26-s − 31-s + 0.999·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(0.512539\)
Root analytic conductor: \(0.715918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1027,\ (\ :0),\ 0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.274042314\)
\(L(\frac12)\) \(\approx\) \(1.274042314\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 - T \)
79 \( 1 - T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.46615755009783933935676078390, −9.335569883374458015250617223554, −8.391963279391228760465524712718, −7.53059444797042939960776823323, −7.10612424694351373910820587978, −6.03885496050121979241949048776, −5.15060879542347511279056135005, −4.39647740261656078063708037035, −3.38756120392712418852395697780, −1.72711042807289924545237413814, 1.22494467818441012398716771442, 2.94938555094551187344708115777, 3.64694614704696204880925619742, 4.18211833303512092634227166706, 5.59579701489961261337259153152, 6.52686038307702376327906944098, 7.56844212499678880318083819740, 8.380450659643299525304038593209, 9.063305693524270150017827572523, 10.27110015600269272082278324988

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