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Properties

Label 2-2527-133.116-c0-0-0
Degree $2$
Conductor $2527$
Sign $0.823 + 0.566i$
Analytic cond. $1.26113$
Root an. cond. $1.12300$
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  + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.5 + 0.866i)7-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (0.173 − 0.984i)16-s + (1.53 + 1.28i)17-s − 20-s + (−0.173 − 0.984i)23-s + (0.173 + 0.984i)28-s + (0.939 − 0.342i)35-s + (0.173 − 0.984i)36-s + (0.939 − 0.342i)43-s + (0.939 + 0.342i)44-s − 45-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.5 + 0.866i)7-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (0.173 − 0.984i)16-s + (1.53 + 1.28i)17-s − 20-s + (−0.173 − 0.984i)23-s + (0.173 + 0.984i)28-s + (0.939 − 0.342i)35-s + (0.173 − 0.984i)36-s + (0.939 − 0.342i)43-s + (0.939 + 0.342i)44-s − 45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $0.823 + 0.566i$
Analytic conductor: \(1.26113\)
Root analytic conductor: \(1.12300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2527,\ (\ :0),\ 0.823 + 0.566i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.357919790\)
\(L(\frac12)\) \(\approx\) \(1.357919790\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 \)
good2 \( 1 + (-0.766 + 0.642i)T^{2} \)
3 \( 1 + (-0.766 + 0.642i)T^{2} \)
5 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T^{2} \)
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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

−9.119841682567213601908361227092, −8.198980974048363487877312849462, −7.50057709395535201062500141271, −6.55537043599220630271648786713, −6.10772400176682353930120211967, −5.13484165638194165665955068313, −4.20913736632022830030144395997, −3.37341032603717984616127972456, −2.11687177419694173041321358207, −1.11283704978368976633298781477, 1.27429923545212349359896446182, 2.79914534185662231936212197561, 3.49339658551966213816577910494, 4.00401978426020820571211216040, 5.28311189720530000333307748641, 6.35336281666760246595559919262, 7.10976539897780340977011292990, 7.58118283127140995910096308788, 7.957381664619089528055225725327, 9.215344345371749373896579706390

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