Properties

Label 2-572-13.12-c1-0-2
Degree $2$
Conductor $572$
Sign $0.0390 - 0.999i$
Analytic cond. $4.56744$
Root an. cond. $2.13715$
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  − 3.32·3-s + 2.74i·5-s − 4.32i·7-s + 8.07·9-s i·11-s + (−3.60 − 0.140i)13-s − 9.12i·15-s + 2.28·17-s + 5.78i·19-s + 14.3i·21-s + 1.58·23-s − 2.52·25-s − 16.8·27-s − 1.44·29-s + 8.74i·31-s + ⋯
L(s)  = 1  − 1.92·3-s + 1.22i·5-s − 1.63i·7-s + 2.69·9-s − 0.301i·11-s + (−0.999 − 0.0390i)13-s − 2.35i·15-s + 0.553·17-s + 1.32i·19-s + 3.14i·21-s + 0.330·23-s − 0.505·25-s − 3.24·27-s − 0.269·29-s + 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0390 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0390 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.0390 - 0.999i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.0390 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391142 + 0.376173i\)
\(L(\frac12)\) \(\approx\) \(0.391142 + 0.376173i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + iT \)
13 \( 1 + (3.60 + 0.140i)T \)
good3 \( 1 + 3.32T + 3T^{2} \)
5 \( 1 - 2.74iT - 5T^{2} \)
7 \( 1 + 4.32iT - 7T^{2} \)
17 \( 1 - 2.28T + 17T^{2} \)
19 \( 1 - 5.78iT - 19T^{2} \)
23 \( 1 - 1.58T + 23T^{2} \)
29 \( 1 + 1.44T + 29T^{2} \)
31 \( 1 - 8.74iT - 31T^{2} \)
37 \( 1 - 7.39iT - 37T^{2} \)
41 \( 1 - 4.33iT - 41T^{2} \)
43 \( 1 + 3.44T + 43T^{2} \)
47 \( 1 - 2.65iT - 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 - 0.986iT - 59T^{2} \)
61 \( 1 + 5.44T + 61T^{2} \)
67 \( 1 + 0.180iT - 67T^{2} \)
71 \( 1 - 15.1iT - 71T^{2} \)
73 \( 1 - 6.72iT - 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + 11.5iT - 83T^{2} \)
89 \( 1 + 6.56iT - 89T^{2} \)
97 \( 1 - 5.25iT - 97T^{2} \)
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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

−10.83102270874798983477269884846, −10.24202278598291193275352991887, −9.990509787064442955871382270883, −7.77695641179923893274000784291, −7.03242864714680645597307191413, −6.58460433767050360287999013564, −5.50551816034581986911931971407, −4.50503015871190510782457207083, −3.43024853924169661377162810687, −1.19924437337209641535381475414, 0.45659696260437373711446877668, 2.11485173961056099699713367907, 4.40346387346905002073277618754, 5.27658746744576742797012116251, 5.50923978957009018022002907142, 6.64888939083563307422546427163, 7.70830087589164313563825106076, 9.109641560957495118621290943854, 9.534119188524100486596843129087, 10.71395949996138924858541104553

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