Properties

Label 2-570-15.2-c1-0-31
Degree $2$
Conductor $570$
Sign $-0.508 + 0.860i$
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 + (1.07 + 1.35i)3-s − 1.00i·4-s + (−1.25 − 1.85i)5-s + (1.72 + 0.194i)6-s + (−3.56 − 3.56i)7-s + (−0.707 − 0.707i)8-s + (−0.668 + 2.92i)9-s + (−2.19 − 0.421i)10-s − 3.35i·11-s + (1.35 − 1.07i)12-s + (0.759 − 0.759i)13-s − 5.03·14-s + (1.15 − 3.69i)15-s − 1.00·16-s + (−0.534 + 0.534i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.623 + 0.781i)3-s − 0.500i·4-s + (−0.561 − 0.827i)5-s + (0.702 + 0.0793i)6-s + (−1.34 − 1.34i)7-s + (−0.250 − 0.250i)8-s + (−0.222 + 0.974i)9-s + (−0.694 − 0.133i)10-s − 1.01i·11-s + (0.390 − 0.311i)12-s + (0.210 − 0.210i)13-s − 1.34·14-s + (0.297 − 0.954i)15-s − 0.250·16-s + (−0.129 + 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.508 + 0.860i$
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: \((2,\ 570,\ (\ :1/2),\ -0.508 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735720 - 1.28954i\)
\(L(\frac12)\) \(\approx\) \(0.735720 - 1.28954i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 - 1.35i)T \)
5 \( 1 + (1.25 + 1.85i)T \)
19 \( 1 - iT \)
good7 \( 1 + (3.56 + 3.56i)T + 7iT^{2} \)
11 \( 1 + 3.35iT - 11T^{2} \)
13 \( 1 + (-0.759 + 0.759i)T - 13iT^{2} \)
17 \( 1 + (0.534 - 0.534i)T - 17iT^{2} \)
23 \( 1 + (3.95 + 3.95i)T + 23iT^{2} \)
29 \( 1 - 3.01T + 29T^{2} \)
31 \( 1 - 8.18T + 31T^{2} \)
37 \( 1 + (-5.57 - 5.57i)T + 37iT^{2} \)
41 \( 1 + 8.43iT - 41T^{2} \)
43 \( 1 + (-4.29 + 4.29i)T - 43iT^{2} \)
47 \( 1 + (-8.00 + 8.00i)T - 47iT^{2} \)
53 \( 1 + (7.04 + 7.04i)T + 53iT^{2} \)
59 \( 1 - 4.32T + 59T^{2} \)
61 \( 1 - 1.35T + 61T^{2} \)
67 \( 1 + (-5.23 - 5.23i)T + 67iT^{2} \)
71 \( 1 - 4.74iT - 71T^{2} \)
73 \( 1 + (5.12 - 5.12i)T - 73iT^{2} \)
79 \( 1 - 0.256iT - 79T^{2} \)
83 \( 1 + (-4.73 - 4.73i)T + 83iT^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 + (9.73 + 9.73i)T + 97iT^{2} \)
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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

−10.31834776877042748814377551100, −9.852666399820267048593484096525, −8.748257621871085061222235558686, −8.045775586334279368157161716105, −6.74319189827278683434156202832, −5.59934460738670166232966309513, −4.26068946848600860340001495937, −3.85603739412586288468313111727, −2.86782998178086647426336846028, −0.65307412465318141129909232787, 2.44708379278634674940502256775, 3.07141378127184456751398140233, 4.28161621087169612694740329927, 6.02911304732169419510451383927, 6.42680490156580114926678262717, 7.36271681126443244558411928638, 8.125921538747833671662502679268, 9.226364739493371691536729380707, 9.827412717372181085158974754675, 11.40308510887591673826269458233

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