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Properties

Label 2-3060-85.4-c1-0-42
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: $\chi_{3060} (2809, \cdot )$
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

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

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