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Properties

Label 2-3060-15.8-c1-0-20
Degree $2$
Conductor $3060$
Sign $0.501 + 0.865i$
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  + (−2.12 − 0.707i)5-s + (2 − 2i)7-s − 2.82i·11-s + (3 + 3i)13-s + (−0.707 − 0.707i)17-s + 4i·19-s + (3.99 + 3i)25-s + 7.07·29-s + (−5.65 + 2.82i)35-s + (1 − i)37-s − 7.07i·41-s + (4 + 4i)43-s + (−2.82 − 2.82i)47-s − i·49-s + (7.07 − 7.07i)53-s + ⋯
L(s)  = 1  + (−0.948 − 0.316i)5-s + (0.755 − 0.755i)7-s − 0.852i·11-s + (0.832 + 0.832i)13-s + (−0.171 − 0.171i)17-s + 0.917i·19-s + (0.799 + 0.600i)25-s + 1.31·29-s + (−0.956 + 0.478i)35-s + (0.164 − 0.164i)37-s − 1.10i·41-s + (0.609 + 0.609i)43-s + (−0.412 − 0.412i)47-s − 0.142i·49-s + (0.971 − 0.971i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.501 + 0.865i$
Analytic conductor: \(24.4342\)
Root analytic conductor: \(4.94309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :1/2),\ 0.501 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.667967335\)
\(L(\frac12)\) \(\approx\) \(1.667967335\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 + 0.707i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-2 + 2i)T - 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 - 7.07T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 + 7.07iT - 41T^{2} \)
43 \( 1 + (-4 - 4i)T + 43iT^{2} \)
47 \( 1 + (2.82 + 2.82i)T + 47iT^{2} \)
53 \( 1 + (-7.07 + 7.07i)T - 53iT^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + (-4 + 4i)T - 67iT^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 + 16iT - 79T^{2} \)
83 \( 1 + (2.82 - 2.82i)T - 83iT^{2} \)
89 \( 1 + 9.89T + 89T^{2} \)
97 \( 1 + (-7 + 7i)T - 97iT^{2} \)
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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

−8.494992431319923720022501478441, −7.946318288441165862055064347849, −7.19698623259019475677496231250, −6.39968924579600577820600613632, −5.46992457113591420839221589380, −4.46314918521144048886711114245, −4.00235929440911112916264515594, −3.14112312636089682738073086195, −1.64153146038753571240599965746, −0.67033532155085308289916729493, 1.02002415199494668980626826559, 2.37641950272014535199230403190, 3.15647022772278214409298145686, 4.27408833413453565298402678813, 4.83644855746088462028831789794, 5.77432336654005273476928172637, 6.67624197784515055100676582603, 7.40241082098405256444446042564, 8.228944884840597198618670969553, 8.551750877201552921536275402069

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