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Properties

Label 2-3060-15.2-c1-0-10
Degree $2$
Conductor $3060$
Sign $0.533 - 0.845i$
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  + (1.30 + 1.81i)5-s + (−3.12 − 3.12i)7-s + 5.89i·11-s + (4.65 − 4.65i)13-s + (−0.707 + 0.707i)17-s − 1.39i·19-s + (6.38 + 6.38i)23-s + (−1.58 + 4.74i)25-s − 3.28·29-s − 3.63·31-s + (1.58 − 9.74i)35-s + (−0.354 − 0.354i)37-s − 5.17i·41-s + (5.26 − 5.26i)43-s + (−7.36 + 7.36i)47-s + ⋯
L(s)  = 1  + (0.584 + 0.811i)5-s + (−1.17 − 1.17i)7-s + 1.77i·11-s + (1.29 − 1.29i)13-s + (−0.171 + 0.171i)17-s − 0.320i·19-s + (1.33 + 1.33i)23-s + (−0.316 + 0.948i)25-s − 0.609·29-s − 0.652·31-s + (0.267 − 1.64i)35-s + (−0.0582 − 0.0582i)37-s − 0.807i·41-s + (0.802 − 0.802i)43-s + (−1.07 + 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(24.4342\)
Root analytic conductor: \(4.94309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :1/2),\ 0.533 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.716105446\)
\(L(\frac12)\) \(\approx\) \(1.716105446\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 - 1.81i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (3.12 + 3.12i)T + 7iT^{2} \)
11 \( 1 - 5.89iT - 11T^{2} \)
13 \( 1 + (-4.65 + 4.65i)T - 13iT^{2} \)
19 \( 1 + 1.39iT - 19T^{2} \)
23 \( 1 + (-6.38 - 6.38i)T + 23iT^{2} \)
29 \( 1 + 3.28T + 29T^{2} \)
31 \( 1 + 3.63T + 31T^{2} \)
37 \( 1 + (0.354 + 0.354i)T + 37iT^{2} \)
41 \( 1 + 5.17iT - 41T^{2} \)
43 \( 1 + (-5.26 + 5.26i)T - 43iT^{2} \)
47 \( 1 + (7.36 - 7.36i)T - 47iT^{2} \)
53 \( 1 + (3.59 + 3.59i)T + 53iT^{2} \)
59 \( 1 - 0.694T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 + (-10.7 - 10.7i)T + 67iT^{2} \)
71 \( 1 - 9.55iT - 71T^{2} \)
73 \( 1 + (-4.25 + 4.25i)T - 73iT^{2} \)
79 \( 1 - 3.03iT - 79T^{2} \)
83 \( 1 + (-1.38 - 1.38i)T + 83iT^{2} \)
89 \( 1 - 9.39T + 89T^{2} \)
97 \( 1 + (-10.1 - 10.1i)T + 97iT^{2} \)
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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

−9.104480790173188569228778158464, −7.80814589403170477772392382975, −7.12470117535421663597051065822, −6.81162051562051751065547940963, −5.87775206157745348125879535358, −5.09206872334232778267043630113, −3.76655448369313105248652650092, −3.44049015732816223914600037720, −2.29365704631660621425265634512, −1.06667707441280092479086506589, 0.62119089319173845776317788449, 1.88637335256557597715308861172, 2.97435364538377806154403034267, 3.69437725880528093701689177448, 4.85444050583409543979169515185, 5.74449941569441532844381970649, 6.23530131783010935256505729601, 6.69679008954985690683853626183, 8.275019352318454811493715742738, 8.730736931131002535196663694950

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