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Properties

Label 2-570-15.8-c1-0-26
Degree $2$
Conductor $570$
Sign $-0.999 - 0.0300i$
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 + (−1.32 + 1.11i)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 + (−1.11 − 1.32i)12-s + (0.342 + 0.342i)13-s − 1.84·14-s + (3.74 + 1.00i)15-s − 1.00·16-s + (−4.25 − 4.25i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.767 + 0.641i)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 + (−0.320 − 0.383i)12-s + (0.0948 + 0.0948i)13-s − 0.493·14-s + (0.965 + 0.258i)15-s − 0.250·16-s + (−1.03 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.999 - 0.0300i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.999 - 0.0300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00222615 + 0.148219i\)
\(L(\frac12)\) \(\approx\) \(0.00222615 + 0.148219i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 1.11i)T \)
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

−10.45302925063636871164929103395, −9.307302691889436499044961437688, −8.999261667984268286886484129700, −7.53072225501612259289000029574, −6.99137651003287103234865448265, −5.26344618449509772039303731003, −4.55729901683102420762339889080, −3.77749638982296788774024434890, −1.77854682414320045655285250190, −0.10828399579358373558186295993, 1.77899572502617606595004104708, 3.42253982879995245195386183109, 5.02261041818851525760570877632, 6.01491766406405828448591877971, 6.59948255459293009151976791934, 7.68815846771993133802969864156, 8.243350520156987019154448276259, 9.242238067284220418833125080922, 10.69054826363325613198910195738, 11.22007969503999983332353279318

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