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Properties

Label 2-3060-85.64-c1-0-41
Degree $2$
Conductor $3060$
Sign $-0.902 + 0.429i$
Analytic cond. $24.4342$
Root an. cond. $4.94309$
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  + (1.18 − 1.89i)5-s + (−1.34 − 1.34i)7-s + (−0.474 + 0.474i)11-s − 7.07i·13-s + (−0.381 − 4.10i)17-s + 1.01i·19-s + (4.42 + 4.42i)23-s + (−2.21 − 4.48i)25-s + (0.737 + 0.737i)29-s + (−0.932 − 0.932i)31-s + (−4.12 + 0.962i)35-s + (3.37 − 3.37i)37-s + (−6.22 + 6.22i)41-s − 8.46·43-s + 11.7i·47-s + ⋯
L(s)  = 1  + (0.528 − 0.849i)5-s + (−0.506 − 0.506i)7-s + (−0.142 + 0.142i)11-s − 1.96i·13-s + (−0.0926 − 0.995i)17-s + 0.232i·19-s + (0.923 + 0.923i)23-s + (−0.442 − 0.896i)25-s + (0.136 + 0.136i)29-s + (−0.167 − 0.167i)31-s + (−0.698 + 0.162i)35-s + (0.554 − 0.554i)37-s + (−0.972 + 0.972i)41-s − 1.29·43-s + 1.71i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.902 + 0.429i$
Analytic conductor: \(24.4342\)
Root analytic conductor: \(4.94309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :1/2),\ -0.902 + 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.263088272\)
\(L(\frac12)\) \(\approx\) \(1.263088272\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-1.18 + 1.89i)T \)
17 \( 1 + (0.381 + 4.10i)T \)
good7 \( 1 + (1.34 + 1.34i)T + 7iT^{2} \)
11 \( 1 + (0.474 - 0.474i)T - 11iT^{2} \)
13 \( 1 + 7.07iT - 13T^{2} \)
19 \( 1 - 1.01iT - 19T^{2} \)
23 \( 1 + (-4.42 - 4.42i)T + 23iT^{2} \)
29 \( 1 + (-0.737 - 0.737i)T + 29iT^{2} \)
31 \( 1 + (0.932 + 0.932i)T + 31iT^{2} \)
37 \( 1 + (-3.37 + 3.37i)T - 37iT^{2} \)
41 \( 1 + (6.22 - 6.22i)T - 41iT^{2} \)
43 \( 1 + 8.46T + 43T^{2} \)
47 \( 1 - 11.7iT - 47T^{2} \)
53 \( 1 + 2.30T + 53T^{2} \)
59 \( 1 + 3.70iT - 59T^{2} \)
61 \( 1 + (-0.879 + 0.879i)T - 61iT^{2} \)
67 \( 1 + 8.74iT - 67T^{2} \)
71 \( 1 + (4.40 + 4.40i)T + 71iT^{2} \)
73 \( 1 + (2.32 - 2.32i)T - 73iT^{2} \)
79 \( 1 + (5.66 - 5.66i)T - 79iT^{2} \)
83 \( 1 - 6.73T + 83T^{2} \)
89 \( 1 - 2.91T + 89T^{2} \)
97 \( 1 + (3.18 - 3.18i)T - 97iT^{2} \)
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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

−8.283143284735671504718259383375, −7.76732649196407913534899542001, −6.91410392270294928370243496733, −5.98500294521059781390923054432, −5.26298728749194480659755181571, −4.72063535772136442638086379844, −3.45986524741549272984150281438, −2.79651220010369023843722826740, −1.38738079046251857493774771474, −0.38765911464423512329044515580, 1.67529659108481876985907132482, 2.46346819339986922170612156387, 3.38802255387758881704352866243, 4.30755398590497442690783873686, 5.28924026833712648471835288664, 6.25971541838102769715116494099, 6.65526754678136901020596675757, 7.26557320485733041663084113983, 8.589118257652206368322491573788, 8.908291057697869915356078233053

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