Properties

Label 2-570-95.37-c1-0-5
Degree $2$
Conductor $570$
Sign $0.919 + 0.393i$
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 + (−0.707 − 0.707i)3-s − 1.00i·4-s + (0.528 + 2.17i)5-s − 1.00·6-s + (0.904 + 0.904i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (1.90 + 1.16i)10-s + 2.66·11-s + (−0.707 + 0.707i)12-s + (0.143 + 0.143i)13-s + 1.27·14-s + (1.16 − 1.90i)15-s − 1.00·16-s + (3.29 + 3.29i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (0.236 + 0.971i)5-s − 0.408·6-s + (0.341 + 0.341i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.603 + 0.367i)10-s + 0.802·11-s + (−0.204 + 0.204i)12-s + (0.0398 + 0.0398i)13-s + 0.341·14-s + (0.300 − 0.493i)15-s − 0.250·16-s + (0.799 + 0.799i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.919 + 0.393i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.919 + 0.393i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84135 - 0.377404i\)
\(L(\frac12)\) \(\approx\) \(1.84135 - 0.377404i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.528 - 2.17i)T \)
19 \( 1 + (-4.05 + 1.59i)T \)
good7 \( 1 + (-0.904 - 0.904i)T + 7iT^{2} \)
11 \( 1 - 2.66T + 11T^{2} \)
13 \( 1 + (-0.143 - 0.143i)T + 13iT^{2} \)
17 \( 1 + (-3.29 - 3.29i)T + 17iT^{2} \)
23 \( 1 + (-1.75 + 1.75i)T - 23iT^{2} \)
29 \( 1 - 2.17T + 29T^{2} \)
31 \( 1 + 2.62iT - 31T^{2} \)
37 \( 1 + (0.984 - 0.984i)T - 37iT^{2} \)
41 \( 1 + 3.74iT - 41T^{2} \)
43 \( 1 + (2.01 - 2.01i)T - 43iT^{2} \)
47 \( 1 + (-3.24 - 3.24i)T + 47iT^{2} \)
53 \( 1 + (2.54 + 2.54i)T + 53iT^{2} \)
59 \( 1 - 2.19T + 59T^{2} \)
61 \( 1 + 8.90T + 61T^{2} \)
67 \( 1 + (4.50 - 4.50i)T - 67iT^{2} \)
71 \( 1 - 2.55iT - 71T^{2} \)
73 \( 1 + (5.25 - 5.25i)T - 73iT^{2} \)
79 \( 1 - 4.22T + 79T^{2} \)
83 \( 1 + (-6.55 + 6.55i)T - 83iT^{2} \)
89 \( 1 - 4.94T + 89T^{2} \)
97 \( 1 + (11.8 - 11.8i)T - 97iT^{2} \)
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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

−10.83329556570078373280711029262, −10.09047398127685588403766588517, −9.115329336580309428186158593856, −7.85868480559283408897694564047, −6.84747155766038526284114738338, −6.07881318561331620899101621623, −5.19917521066221735508032589086, −3.85311957704317363610529591962, −2.73735879341228160890266913760, −1.45336961431069681295976527545, 1.22170529453993031958503762319, 3.33181489892417944509203766947, 4.44816896727005505075247525849, 5.18310145283099662955469974824, 5.98669092662067251657664770152, 7.14697688274713886819664976560, 8.050270677037142114470705101644, 9.119489559078163381143602138960, 9.729809812859201681373016325997, 10.89268817071902795863591689455

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