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Properties

Label 2-3060-85.64-c1-0-0
Degree $2$
Conductor $3060$
Sign $-0.684 + 0.729i$
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  + (−1.60 + 1.56i)5-s + (−0.816 − 0.816i)7-s + (−3.94 + 3.94i)11-s + 1.15i·13-s + (2.67 + 3.13i)17-s + 6.31i·19-s + (2.66 + 2.66i)23-s + (0.120 − 4.99i)25-s + (−5.68 − 5.68i)29-s + (−4.61 − 4.61i)31-s + (2.58 + 0.0311i)35-s + (1.62 − 1.62i)37-s + (8.22 − 8.22i)41-s + 6.10·43-s + 10.9i·47-s + ⋯
L(s)  = 1  + (−0.715 + 0.698i)5-s + (−0.308 − 0.308i)7-s + (−1.18 + 1.18i)11-s + 0.320i·13-s + (0.649 + 0.760i)17-s + 1.44i·19-s + (0.554 + 0.554i)23-s + (0.0241 − 0.999i)25-s + (−1.05 − 1.05i)29-s + (−0.829 − 0.829i)31-s + (0.436 + 0.00526i)35-s + (0.267 − 0.267i)37-s + (1.28 − 1.28i)41-s + 0.931·43-s + 1.59i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.684 + 0.729i$
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: \((2,\ 3060,\ (\ :1/2),\ -0.684 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08064782349\)
\(L(\frac12)\) \(\approx\) \(0.08064782349\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 - 1.56i)T \)
17 \( 1 + (-2.67 - 3.13i)T \)
good7 \( 1 + (0.816 + 0.816i)T + 7iT^{2} \)
11 \( 1 + (3.94 - 3.94i)T - 11iT^{2} \)
13 \( 1 - 1.15iT - 13T^{2} \)
19 \( 1 - 6.31iT - 19T^{2} \)
23 \( 1 + (-2.66 - 2.66i)T + 23iT^{2} \)
29 \( 1 + (5.68 + 5.68i)T + 29iT^{2} \)
31 \( 1 + (4.61 + 4.61i)T + 31iT^{2} \)
37 \( 1 + (-1.62 + 1.62i)T - 37iT^{2} \)
41 \( 1 + (-8.22 + 8.22i)T - 41iT^{2} \)
43 \( 1 - 6.10T + 43T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 + 9.43T + 53T^{2} \)
59 \( 1 + 5.58iT - 59T^{2} \)
61 \( 1 + (1.64 - 1.64i)T - 61iT^{2} \)
67 \( 1 + 5.66iT - 67T^{2} \)
71 \( 1 + (5.49 + 5.49i)T + 71iT^{2} \)
73 \( 1 + (10.7 - 10.7i)T - 73iT^{2} \)
79 \( 1 + (-2.42 + 2.42i)T - 79iT^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + (6.19 - 6.19i)T - 97iT^{2} \)
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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

−9.426019705773666937438742632586, −8.052330651455248570508493833229, −7.66216374368175502471064032688, −7.21730195147571790821309175485, −6.11164738474174954457002338107, −5.49903401280001443953352286919, −4.24579469238115675956788721625, −3.80856590649470393463926183917, −2.74580076995349286147434264744, −1.77374217896446162251828645253, 0.02859525812045974161894913560, 1.04472015852361170246691582870, 2.80289869045447876727525354670, 3.18448867058097579824832788818, 4.42716422804136065858475085628, 5.25472898984234399802507645466, 5.66769647197016920288475663814, 6.89133361596676217142610091894, 7.58765727053431805704351061571, 8.240412977815686220552871890451

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