Properties

Label 2-570-15.2-c1-0-18
Degree $2$
Conductor $570$
Sign $0.989 + 0.147i$
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 + (1.11 + 1.32i)3-s − 1.00i·4-s + (1.28 − 1.82i)5-s + (1.72 + 0.154i)6-s + (1.30 + 1.30i)7-s + (−0.707 − 0.707i)8-s + (−0.531 + 2.95i)9-s + (−0.384 − 2.20i)10-s + 5.15i·11-s + (1.32 − 1.11i)12-s + (0.342 − 0.342i)13-s + 1.84·14-s + (3.85 − 0.324i)15-s − 1.00·16-s + (4.25 − 4.25i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.641 + 0.767i)3-s − 0.500i·4-s + (0.574 − 0.818i)5-s + (0.704 + 0.0628i)6-s + (0.493 + 0.493i)7-s + (−0.250 − 0.250i)8-s + (−0.177 + 0.984i)9-s + (−0.121 − 0.696i)10-s + 1.55i·11-s + (0.383 − 0.320i)12-s + (0.0948 − 0.0948i)13-s + 0.493·14-s + (0.996 − 0.0836i)15-s − 0.250·16-s + (1.03 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.989 + 0.147i$
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: \((2,\ 570,\ (\ :1/2),\ 0.989 + 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60675 - 0.193292i\)
\(L(\frac12)\) \(\approx\) \(2.60675 - 0.193292i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 - 1.32i)T \)
5 \( 1 + (-1.28 + 1.82i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-1.30 - 1.30i)T + 7iT^{2} \)
11 \( 1 - 5.15iT - 11T^{2} \)
13 \( 1 + (-0.342 + 0.342i)T - 13iT^{2} \)
17 \( 1 + (-4.25 + 4.25i)T - 17iT^{2} \)
23 \( 1 + (3.32 + 3.32i)T + 23iT^{2} \)
29 \( 1 - 8.73T + 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 + (-0.317 - 0.317i)T + 37iT^{2} \)
41 \( 1 + 8.89iT - 41T^{2} \)
43 \( 1 + (8.84 - 8.84i)T - 43iT^{2} \)
47 \( 1 + (-2.14 + 2.14i)T - 47iT^{2} \)
53 \( 1 + (-8.17 - 8.17i)T + 53iT^{2} \)
59 \( 1 + 1.27T + 59T^{2} \)
61 \( 1 + 6.26T + 61T^{2} \)
67 \( 1 + (1.50 + 1.50i)T + 67iT^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 + (2.73 - 2.73i)T - 73iT^{2} \)
79 \( 1 + 5.23iT - 79T^{2} \)
83 \( 1 + (9.88 + 9.88i)T + 83iT^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 + (-4.02 - 4.02i)T + 97iT^{2} \)
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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

−10.44931879480081191019836975016, −9.897666249462857933241969281443, −9.171454976269727690411042785966, −8.353785727059349044826190930385, −7.21420407749963689370405675480, −5.61345181440522141493143142649, −4.93191951927827299971575741416, −4.22831453066714226904057413254, −2.73922130437948488240161876311, −1.76515683998109660919010546945, 1.57833893372132061819714272851, 3.09300471637972016539078343835, 3.77613120047496067475619601766, 5.60193955823044883734925041346, 6.19600071304275084729075310139, 7.14322657823356269768314279502, 8.003827956920449098773589832477, 8.614756774428673811579572077798, 9.875889670400491894716565631824, 10.84596089715842795557502315928

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