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Properties

Label 2-572-13.12-c1-0-12
Degree $2$
Conductor $572$
Sign $0.435 + 0.900i$
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  + 2.51·3-s − 3.81i·5-s − 1.51i·7-s + 3.30·9-s + i·11-s + (−3.24 + 1.56i)13-s − 9.58i·15-s + 5.13·17-s + 1.83i·19-s − 3.79i·21-s − 5.32·23-s − 9.55·25-s + 0.765·27-s + 9.51·29-s − 9.81i·31-s + ⋯
L(s)  = 1  + 1.44·3-s − 1.70i·5-s − 0.571i·7-s + 1.10·9-s + 0.301i·11-s + (−0.900 + 0.435i)13-s − 2.47i·15-s + 1.24·17-s + 0.420i·19-s − 0.827i·21-s − 1.11·23-s − 1.91·25-s + 0.147·27-s + 1.76·29-s − 1.76i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.435 + 0.900i$
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: \((2,\ 572,\ (\ :1/2),\ 0.435 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91016 - 1.19807i\)
\(L(\frac12)\) \(\approx\) \(1.91016 - 1.19807i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.24 - 1.56i)T \)
good3 \( 1 - 2.51T + 3T^{2} \)
5 \( 1 + 3.81iT - 5T^{2} \)
7 \( 1 + 1.51iT - 7T^{2} \)
17 \( 1 - 5.13T + 17T^{2} \)
19 \( 1 - 1.83iT - 19T^{2} \)
23 \( 1 + 5.32T + 23T^{2} \)
29 \( 1 - 9.51T + 29T^{2} \)
31 \( 1 + 9.81iT - 31T^{2} \)
37 \( 1 - 3.20iT - 37T^{2} \)
41 \( 1 - 6.74iT - 41T^{2} \)
43 \( 1 - 7.51T + 43T^{2} \)
47 \( 1 - 9.02iT - 47T^{2} \)
53 \( 1 + 1.06T + 53T^{2} \)
59 \( 1 - 8.19iT - 59T^{2} \)
61 \( 1 - 5.51T + 61T^{2} \)
67 \( 1 + 4.46iT - 67T^{2} \)
71 \( 1 - 3.58iT - 71T^{2} \)
73 \( 1 - 9.71iT - 73T^{2} \)
79 \( 1 + 4.25T + 79T^{2} \)
83 \( 1 - 4.98iT - 83T^{2} \)
89 \( 1 + 17.8iT - 89T^{2} \)
97 \( 1 + 4.18iT - 97T^{2} \)
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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

−9.923138813304545241635532942770, −9.749581031167250377692387744268, −8.741469017018877184776480906376, −7.994904109432549819991561925956, −7.52407271577268965858796272454, −5.88703083006803515669113440783, −4.60263593682069833882933190920, −4.00999326514715338167956485176, −2.53876264786716688172336186594, −1.21841841103577539839715012567, 2.30598217705380690069498805466, 2.93051507855104843744430943545, 3.71765883125769909605392259340, 5.42169995294173769580496885465, 6.62315922964251077167821845098, 7.46607479582212819923517461901, 8.157695190420697274276893274163, 9.098413874446420751654198439972, 10.13204507453179728966666946026, 10.48898279581817970661509932221

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