Properties

Label 2-570-15.2-c1-0-11
Degree $2$
Conductor $570$
Sign $0.955 - 0.295i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
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  + (0.707 − 0.707i)2-s + (−1.33 + 1.10i)3-s − 1.00i·4-s + (2.20 + 0.356i)5-s + (−0.162 + 1.72i)6-s + (1.29 + 1.29i)7-s + (−0.707 − 0.707i)8-s + (0.558 − 2.94i)9-s + (1.81 − 1.30i)10-s + 4.05i·11-s + (1.10 + 1.33i)12-s + (0.882 − 0.882i)13-s + 1.83·14-s + (−3.33 + 1.96i)15-s − 1.00·16-s + (−1.10 + 1.10i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.770 + 0.637i)3-s − 0.500i·4-s + (0.987 + 0.159i)5-s + (−0.0661 + 0.704i)6-s + (0.489 + 0.489i)7-s + (−0.250 − 0.250i)8-s + (0.186 − 0.982i)9-s + (0.573 − 0.413i)10-s + 1.22i·11-s + (0.318 + 0.385i)12-s + (0.244 − 0.244i)13-s + 0.489·14-s + (−0.861 + 0.506i)15-s − 0.250·16-s + (−0.267 + 0.267i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.955 - 0.295i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.955 - 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80252 + 0.272233i\)
\(L(\frac12)\) \(\approx\) \(1.80252 + 0.272233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (1.33 - 1.10i)T \)
5 \( 1 + (-2.20 - 0.356i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-1.29 - 1.29i)T + 7iT^{2} \)
11 \( 1 - 4.05iT - 11T^{2} \)
13 \( 1 + (-0.882 + 0.882i)T - 13iT^{2} \)
17 \( 1 + (1.10 - 1.10i)T - 17iT^{2} \)
23 \( 1 + (0.877 + 0.877i)T + 23iT^{2} \)
29 \( 1 - 5.03T + 29T^{2} \)
31 \( 1 - 9.76T + 31T^{2} \)
37 \( 1 + (-3.02 - 3.02i)T + 37iT^{2} \)
41 \( 1 + 1.87iT - 41T^{2} \)
43 \( 1 + (0.199 - 0.199i)T - 43iT^{2} \)
47 \( 1 + (8.33 - 8.33i)T - 47iT^{2} \)
53 \( 1 + (6.34 + 6.34i)T + 53iT^{2} \)
59 \( 1 + 1.21T + 59T^{2} \)
61 \( 1 - 1.45T + 61T^{2} \)
67 \( 1 + (2.25 + 2.25i)T + 67iT^{2} \)
71 \( 1 + 7.22iT - 71T^{2} \)
73 \( 1 + (3.04 - 3.04i)T - 73iT^{2} \)
79 \( 1 + 13.1iT - 79T^{2} \)
83 \( 1 + (-4.01 - 4.01i)T + 83iT^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + (7.86 + 7.86i)T + 97iT^{2} \)
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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

−10.73069110227696753502164725743, −10.02738205150792943921855304170, −9.511099291384172474497042808331, −8.317522924086869198710799097365, −6.69386716481346893669507078166, −6.05690536179451548930052463087, −5.01957774519316511176677391615, −4.44719907241773925153485531606, −2.88597813822093315592784092231, −1.58181340859402036576220848624, 1.14093584663875428369021021466, 2.70879289219875899022802735142, 4.43041788501955161899358274503, 5.30406592311758858515473805876, 6.19005252034194497565328378589, 6.73035986542705882875435008837, 7.925203208327691187913456593415, 8.663350614025205189688433676389, 9.955194231653985740474323625933, 10.92060976008561616009057975150

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