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Properties

Label 2-3060-17.13-c1-0-22
Degree $2$
Conductor $3060$
Sign $0.865 + 0.500i$
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  + (0.707 + 0.707i)5-s + (1.68 − 1.68i)7-s + (2.85 − 2.85i)11-s + 6.67·13-s + (3.28 + 2.48i)17-s − 1.65i·19-s + (2.78 − 2.78i)23-s + 1.00i·25-s + (0.119 + 0.119i)29-s + (−6.53 − 6.53i)31-s + 2.38·35-s + (−3.20 − 3.20i)37-s + (−8.52 + 8.52i)41-s + 0.294i·43-s − 0.832·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + (0.636 − 0.636i)7-s + (0.860 − 0.860i)11-s + 1.85·13-s + (0.797 + 0.602i)17-s − 0.378i·19-s + (0.579 − 0.579i)23-s + 0.200i·25-s + (0.0222 + 0.0222i)29-s + (−1.17 − 1.17i)31-s + 0.402·35-s + (−0.526 − 0.526i)37-s + (−1.33 + 1.33i)41-s + 0.0449i·43-s − 0.121·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.865 + 0.500i$
Analytic conductor: \(24.4342\)
Root analytic conductor: \(4.94309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :1/2),\ 0.865 + 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.568487781\)
\(L(\frac12)\) \(\approx\) \(2.568487781\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (-3.28 - 2.48i)T \)
good7 \( 1 + (-1.68 + 1.68i)T - 7iT^{2} \)
11 \( 1 + (-2.85 + 2.85i)T - 11iT^{2} \)
13 \( 1 - 6.67T + 13T^{2} \)
19 \( 1 + 1.65iT - 19T^{2} \)
23 \( 1 + (-2.78 + 2.78i)T - 23iT^{2} \)
29 \( 1 + (-0.119 - 0.119i)T + 29iT^{2} \)
31 \( 1 + (6.53 + 6.53i)T + 31iT^{2} \)
37 \( 1 + (3.20 + 3.20i)T + 37iT^{2} \)
41 \( 1 + (8.52 - 8.52i)T - 41iT^{2} \)
43 \( 1 - 0.294iT - 43T^{2} \)
47 \( 1 + 0.832T + 47T^{2} \)
53 \( 1 - 2.84iT - 53T^{2} \)
59 \( 1 - 6.00iT - 59T^{2} \)
61 \( 1 + (-0.781 + 0.781i)T - 61iT^{2} \)
67 \( 1 - 7.21T + 67T^{2} \)
71 \( 1 + (7.77 + 7.77i)T + 71iT^{2} \)
73 \( 1 + (7.75 + 7.75i)T + 73iT^{2} \)
79 \( 1 + (-2.48 + 2.48i)T - 79iT^{2} \)
83 \( 1 + 9.90iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + (-0.805 - 0.805i)T + 97iT^{2} \)
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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

−8.667531732413028725396188954464, −7.973889141128647651078444039443, −7.13641999056521061833273644035, −6.20197996026670520555865914370, −5.88173448344607515633023679941, −4.70590847301164931202201437268, −3.75743647233061010416529854312, −3.26159676723073073119314186290, −1.72779728580975695835192002562, −0.960308336132632009480021196792, 1.29494096690614084314828956965, 1.83607373718543566761341483585, 3.29535008612838786775401581484, 3.97987787561828518126828482808, 5.19141838001909474195763890791, 5.49045491106007609175895048919, 6.56685459900352543044747324330, 7.17597024075079317098339930695, 8.269369138256107344806258255724, 8.734535952439597096088701389248

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