Properties

Label 2-572-11.4-c1-0-3
Degree $2$
Conductor $572$
Sign $-0.670 - 0.742i$
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  + (−0.344 + 1.05i)3-s + (−0.665 + 0.483i)5-s + (0.816 + 2.51i)7-s + (1.42 + 1.03i)9-s + (−3.22 − 0.761i)11-s + (0.809 + 0.587i)13-s + (−0.283 − 0.871i)15-s + (−4.38 + 3.18i)17-s + (1.81 − 5.59i)19-s − 2.94·21-s − 2.27·23-s + (−1.33 + 4.11i)25-s + (−4.28 + 3.11i)27-s + (2.34 + 7.22i)29-s + (−0.936 − 0.680i)31-s + ⋯
L(s)  = 1  + (−0.198 + 0.611i)3-s + (−0.297 + 0.216i)5-s + (0.308 + 0.949i)7-s + (0.474 + 0.344i)9-s + (−0.973 − 0.229i)11-s + (0.224 + 0.163i)13-s + (−0.0730 − 0.224i)15-s + (−1.06 + 0.773i)17-s + (0.417 − 1.28i)19-s − 0.642·21-s − 0.473·23-s + (−0.267 + 0.822i)25-s + (−0.825 + 0.599i)27-s + (0.436 + 1.34i)29-s + (−0.168 − 0.122i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.670 - 0.742i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.670 - 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.416270 + 0.937001i\)
\(L(\frac12)\) \(\approx\) \(0.416270 + 0.937001i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.22 + 0.761i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (0.344 - 1.05i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.665 - 0.483i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.816 - 2.51i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (4.38 - 3.18i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.81 + 5.59i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.27T + 23T^{2} \)
29 \( 1 + (-2.34 - 7.22i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.936 + 0.680i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.433 - 1.33i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.26 - 3.89i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.44T + 43T^{2} \)
47 \( 1 + (1.84 - 5.67i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.23 + 5.25i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.80 - 5.55i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.67 + 6.30i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 3.00T + 67T^{2} \)
71 \( 1 + (-7.89 + 5.73i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.32 - 4.07i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.2 + 8.14i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.92 + 2.85i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + (-2.15 - 1.56i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.07330204664195582768876286874, −10.31686378037707922591723104145, −9.282005047767800887281780119997, −8.510499030700655315636892357380, −7.56891838013486034176110624239, −6.48526796897033237110633922757, −5.28900466773451541730783662839, −4.67923459866239840745968581059, −3.34784889193230842036701418907, −2.05605701684000618770552752674, 0.58741949172446574308107862914, 2.11318418241383746355944106599, 3.80464778210182708453555407066, 4.66421497415565389488933544303, 5.92878026456759296195375507843, 6.95493914043655659215481489368, 7.69102624282009073958049287253, 8.347032766854376056855515446195, 9.772207298632430202609459851008, 10.33966595829073570919151063819

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