Properties

Label 2-570-15.2-c1-0-9
Degree $2$
Conductor $570$
Sign $0.844 + 0.536i$
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 + (−1.63 − 0.577i)3-s − 1.00i·4-s + (−2.23 + 0.0135i)5-s + (−1.56 + 0.746i)6-s + (3.38 + 3.38i)7-s + (−0.707 − 0.707i)8-s + (2.33 + 1.88i)9-s + (−1.57 + 1.59i)10-s + 1.22i·11-s + (−0.577 + 1.63i)12-s + (3.51 − 3.51i)13-s + 4.78·14-s + (3.65 + 1.26i)15-s − 1.00·16-s + (−0.826 + 0.826i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.942 − 0.333i)3-s − 0.500i·4-s + (−0.999 + 0.00606i)5-s + (−0.638 + 0.304i)6-s + (1.27 + 1.27i)7-s + (−0.250 − 0.250i)8-s + (0.777 + 0.628i)9-s + (−0.496 + 0.503i)10-s + 0.369i·11-s + (−0.166 + 0.471i)12-s + (0.975 − 0.975i)13-s + 1.27·14-s + (0.944 + 0.327i)15-s − 0.250·16-s + (−0.200 + 0.200i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.844 + 0.536i$
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: \((2,\ 570,\ (\ :1/2),\ 0.844 + 0.536i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34656 - 0.391404i\)
\(L(\frac12)\) \(\approx\) \(1.34656 - 0.391404i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 + 0.577i)T \)
5 \( 1 + (2.23 - 0.0135i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-3.38 - 3.38i)T + 7iT^{2} \)
11 \( 1 - 1.22iT - 11T^{2} \)
13 \( 1 + (-3.51 + 3.51i)T - 13iT^{2} \)
17 \( 1 + (0.826 - 0.826i)T - 17iT^{2} \)
23 \( 1 + (-1.04 - 1.04i)T + 23iT^{2} \)
29 \( 1 - 8.14T + 29T^{2} \)
31 \( 1 - 7.41T + 31T^{2} \)
37 \( 1 + (5.83 + 5.83i)T + 37iT^{2} \)
41 \( 1 - 3.53iT - 41T^{2} \)
43 \( 1 + (2.68 - 2.68i)T - 43iT^{2} \)
47 \( 1 + (-6.84 + 6.84i)T - 47iT^{2} \)
53 \( 1 + (-3.40 - 3.40i)T + 53iT^{2} \)
59 \( 1 - 9.91T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + (-6.00 - 6.00i)T + 67iT^{2} \)
71 \( 1 - 8.35iT - 71T^{2} \)
73 \( 1 + (3.99 - 3.99i)T - 73iT^{2} \)
79 \( 1 - 6.75iT - 79T^{2} \)
83 \( 1 + (0.717 + 0.717i)T + 83iT^{2} \)
89 \( 1 - 0.384T + 89T^{2} \)
97 \( 1 + (12.5 + 12.5i)T + 97iT^{2} \)
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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

−10.99297341382999966797147204213, −10.25438589979297509471706260394, −8.647563342367019568420058119176, −8.117830994564349738169866912698, −6.92521742538708542362451272481, −5.78532955480646902860082531509, −5.05966657190350221504565036350, −4.21213950802839091905698722353, −2.64853742559141296346331245182, −1.17730119146007592173369824526, 1.05169450331380853135380840907, 3.63230247103950802439977834198, 4.41981481313099477892954796035, 4.92935678683024736238929411351, 6.39674657002658770853714582693, 7.03299187060531984032957034974, 8.011794633090448220500269966374, 8.759323189607538299457262113031, 10.38278528211578586253257930492, 10.94401184598750347054792679427

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