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Properties

Label 2-3060-85.84-c1-0-35
Degree $2$
Conductor $3060$
Sign $-0.125 + 0.992i$
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  + (0.280 − 2.21i)5-s − 0.972i·11-s − 4.43i·13-s + 4.12·17-s + 3.68·19-s + 5.43·23-s + (−4.84 − 1.24i)25-s + 6.92i·29-s − 2.49i·41-s − 5.95i·43-s − 7·49-s + (−2.15 − 0.273i)55-s + (−9.84 − 1.24i)65-s − 14.2i·67-s − 3.46i·71-s + ⋯
L(s)  = 1  + (0.125 − 0.992i)5-s − 0.293i·11-s − 1.23i·13-s + 0.999·17-s + 0.845·19-s + 1.13·23-s + (−0.968 − 0.249i)25-s + 1.28i·29-s − 0.389i·41-s − 0.908i·43-s − 49-s + (−0.290 − 0.0368i)55-s + (−1.22 − 0.154i)65-s − 1.74i·67-s − 0.411i·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.125 + 0.992i$
Analytic conductor: \(24.4342\)
Root analytic conductor: \(4.94309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :1/2),\ -0.125 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.831671864\)
\(L(\frac12)\) \(\approx\) \(1.831671864\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.280 + 2.21i)T \)
17 \( 1 - 4.12T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 0.972iT - 11T^{2} \)
13 \( 1 + 4.43iT - 13T^{2} \)
19 \( 1 - 3.68T + 19T^{2} \)
23 \( 1 - 5.43T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 2.49iT - 41T^{2} \)
43 \( 1 + 5.95iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 14.2iT - 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−8.519794438410747078732161144823, −7.83504873274345566805258552838, −7.16839244142777288374947529540, −6.03943073963704565086414533196, −5.28331798921186072224411488530, −4.94120044784059892486708711586, −3.61031855013224482377899246410, −2.98963963091236903255125839115, −1.52606701279843557085264320391, −0.63018086961991004055976892362, 1.30632601307830693904172939852, 2.44907687803520375042750995159, 3.25697390223354867869450402243, 4.15160245752373455905869672042, 5.09265046837165691782687584438, 6.00946645979842112350501086558, 6.70469579945648307964087404616, 7.36284210673040822199901536534, 7.988889411153853872417108911520, 9.053602425611188705004668276817

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