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Properties

Label 2-620-20.7-c1-0-64
Degree $2$
Conductor $620$
Sign $-0.983 + 0.183i$
Analytic cond. $4.95072$
Root an. cond. $2.22502$
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  + (−1.25 + 0.646i)2-s + (−1.39 − 1.39i)3-s + (1.16 − 1.62i)4-s + (−2.16 + 0.556i)5-s + (2.64 + 0.849i)6-s + (2.15 − 2.15i)7-s + (−0.409 + 2.79i)8-s + 0.871i·9-s + (2.36 − 2.10i)10-s − 5.67i·11-s + (−3.88 + 0.645i)12-s + (−1.47 + 1.47i)13-s + (−1.31 + 4.10i)14-s + (3.78 + 2.23i)15-s + (−1.29 − 3.78i)16-s + (1.62 + 1.62i)17-s + ⋯
L(s)  = 1  + (−0.889 + 0.457i)2-s + (−0.803 − 0.803i)3-s + (0.581 − 0.813i)4-s + (−0.968 + 0.248i)5-s + (1.08 + 0.346i)6-s + (0.814 − 0.814i)7-s + (−0.144 + 0.989i)8-s + 0.290i·9-s + (0.747 − 0.664i)10-s − 1.71i·11-s + (−1.12 + 0.186i)12-s + (−0.410 + 0.410i)13-s + (−0.351 + 1.09i)14-s + (0.977 + 0.578i)15-s + (−0.323 − 0.946i)16-s + (0.394 + 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(620\)    =    \(2^{2} \cdot 5 \cdot 31\)
Sign: $-0.983 + 0.183i$
Analytic conductor: \(4.95072\)
Root analytic conductor: \(2.22502\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{620} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 620,\ (\ :1/2),\ -0.983 + 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0301382 - 0.325879i\)
\(L(\frac12)\) \(\approx\) \(0.0301382 - 0.325879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 - 0.646i)T \)
5 \( 1 + (2.16 - 0.556i)T \)
31 \( 1 + iT \)
good3 \( 1 + (1.39 + 1.39i)T + 3iT^{2} \)
7 \( 1 + (-2.15 + 2.15i)T - 7iT^{2} \)
11 \( 1 + 5.67iT - 11T^{2} \)
13 \( 1 + (1.47 - 1.47i)T - 13iT^{2} \)
17 \( 1 + (-1.62 - 1.62i)T + 17iT^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + (0.674 + 0.674i)T + 23iT^{2} \)
29 \( 1 + 3.16iT - 29T^{2} \)
37 \( 1 + (2.62 + 2.62i)T + 37iT^{2} \)
41 \( 1 + 6.95T + 41T^{2} \)
43 \( 1 + (5.71 + 5.71i)T + 43iT^{2} \)
47 \( 1 + (7.76 - 7.76i)T - 47iT^{2} \)
53 \( 1 + (9.46 - 9.46i)T - 53iT^{2} \)
59 \( 1 + 6.38T + 59T^{2} \)
61 \( 1 + 7.62T + 61T^{2} \)
67 \( 1 + (-8.48 + 8.48i)T - 67iT^{2} \)
71 \( 1 - 0.618iT - 71T^{2} \)
73 \( 1 + (-2.16 + 2.16i)T - 73iT^{2} \)
79 \( 1 - 6.50T + 79T^{2} \)
83 \( 1 + (1.57 + 1.57i)T + 83iT^{2} \)
89 \( 1 - 0.579iT - 89T^{2} \)
97 \( 1 + (-8.85 - 8.85i)T + 97iT^{2} \)
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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

−10.48023678168093052133246578810, −9.155767671854035052035188572615, −8.037067671648708961918044840286, −7.71731183375446287965738168549, −6.74911131751199778153757234816, −6.02847488008029291921104537996, −4.90299352821233740618993507200, −3.38960578283215925337889323591, −1.38307545582879963819205042168, −0.28797334008463380856053413756, 1.80024841258970872685675858243, 3.34646380447767223808252896822, 4.73406327181252089997460582885, 5.12634746531057790920667624517, 6.85448141629701199859295643350, 7.74468799928842641771811570555, 8.413111890811449418947482263587, 9.588417682324865245602658886071, 10.07727831292523100315663733650, 11.05460598024862279661695427993

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