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Properties

Label 2-3060-85.64-c1-0-3
Degree $2$
Conductor $3060$
Sign $-0.978 - 0.206i$
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  + (0.880 + 2.05i)5-s + (−1.19 − 1.19i)7-s + (−0.103 + 0.103i)11-s + 0.176i·13-s + (−4.11 + 0.206i)17-s − 1.74i·19-s + (1.53 + 1.53i)23-s + (−3.45 + 3.61i)25-s + (6.66 + 6.66i)29-s + (−5.47 − 5.47i)31-s + (1.40 − 3.51i)35-s + (−5.73 + 5.73i)37-s + (−0.296 + 0.296i)41-s − 4.71·43-s − 6.04i·47-s + ⋯
L(s)  = 1  + (0.393 + 0.919i)5-s + (−0.452 − 0.452i)7-s + (−0.0311 + 0.0311i)11-s + 0.0490i·13-s + (−0.998 + 0.0501i)17-s − 0.401i·19-s + (0.319 + 0.319i)23-s + (−0.690 + 0.723i)25-s + (1.23 + 1.23i)29-s + (−0.982 − 0.982i)31-s + (0.237 − 0.593i)35-s + (−0.943 + 0.943i)37-s + (−0.0462 + 0.0462i)41-s − 0.719·43-s − 0.881i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.978 - 0.206i$
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: \((2,\ 3060,\ (\ :1/2),\ -0.978 - 0.206i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5187428613\)
\(L(\frac12)\) \(\approx\) \(0.5187428613\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.880 - 2.05i)T \)
17 \( 1 + (4.11 - 0.206i)T \)
good7 \( 1 + (1.19 + 1.19i)T + 7iT^{2} \)
11 \( 1 + (0.103 - 0.103i)T - 11iT^{2} \)
13 \( 1 - 0.176iT - 13T^{2} \)
19 \( 1 + 1.74iT - 19T^{2} \)
23 \( 1 + (-1.53 - 1.53i)T + 23iT^{2} \)
29 \( 1 + (-6.66 - 6.66i)T + 29iT^{2} \)
31 \( 1 + (5.47 + 5.47i)T + 31iT^{2} \)
37 \( 1 + (5.73 - 5.73i)T - 37iT^{2} \)
41 \( 1 + (0.296 - 0.296i)T - 41iT^{2} \)
43 \( 1 + 4.71T + 43T^{2} \)
47 \( 1 + 6.04iT - 47T^{2} \)
53 \( 1 + 7.52T + 53T^{2} \)
59 \( 1 - 12.8iT - 59T^{2} \)
61 \( 1 + (6.00 - 6.00i)T - 61iT^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 + (-2.57 - 2.57i)T + 71iT^{2} \)
73 \( 1 + (0.427 - 0.427i)T - 73iT^{2} \)
79 \( 1 + (-2.74 + 2.74i)T - 79iT^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 + (-9.87 + 9.87i)T - 97iT^{2} \)
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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

−9.041320320785186548301302830609, −8.421779332208781240696843605988, −7.14655869669163343939225640508, −7.00879820998326652820566554331, −6.17656952247140251804309419505, −5.29650874001894349410071097860, −4.33149872134385165611705789425, −3.38108453080053084699922290526, −2.66821843195266711941418055672, −1.55166004286287741725029555624, 0.15195490957721253396558921979, 1.57548742271961830777510159759, 2.51427665173662681674501965660, 3.59343779551508860338813917185, 4.62815574853594160655371914361, 5.19434390828227743431381524681, 6.16528600509789673110531142039, 6.62440385306263666279768387246, 7.78951286585120322772273020981, 8.456806236672871267409081818326

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