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Properties

Label 2-3060-85.64-c1-0-12
Degree $2$
Conductor $3060$
Sign $0.589 - 0.807i$
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  + (2.19 − 0.417i)5-s + (−3.15 − 3.15i)7-s + (−0.848 + 0.848i)11-s + 5.93i·13-s + (4.07 + 0.639i)17-s + 5.87i·19-s + (−5.63 − 5.63i)23-s + (4.65 − 1.83i)25-s + (1.39 + 1.39i)29-s + (5.11 + 5.11i)31-s + (−8.24 − 5.61i)35-s + (−4.95 + 4.95i)37-s + (−1.70 + 1.70i)41-s − 6.12·43-s − 0.483i·47-s + ⋯
L(s)  = 1  + (0.982 − 0.186i)5-s + (−1.19 − 1.19i)7-s + (−0.255 + 0.255i)11-s + 1.64i·13-s + (0.987 + 0.155i)17-s + 1.34i·19-s + (−1.17 − 1.17i)23-s + (0.930 − 0.366i)25-s + (0.259 + 0.259i)29-s + (0.918 + 0.918i)31-s + (−1.39 − 0.948i)35-s + (−0.813 + 0.813i)37-s + (−0.266 + 0.266i)41-s − 0.934·43-s − 0.0705i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.589 - 0.807i$
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: \((2,\ 3060,\ (\ :1/2),\ 0.589 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.592791916\)
\(L(\frac12)\) \(\approx\) \(1.592791916\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 + 0.417i)T \)
17 \( 1 + (-4.07 - 0.639i)T \)
good7 \( 1 + (3.15 + 3.15i)T + 7iT^{2} \)
11 \( 1 + (0.848 - 0.848i)T - 11iT^{2} \)
13 \( 1 - 5.93iT - 13T^{2} \)
19 \( 1 - 5.87iT - 19T^{2} \)
23 \( 1 + (5.63 + 5.63i)T + 23iT^{2} \)
29 \( 1 + (-1.39 - 1.39i)T + 29iT^{2} \)
31 \( 1 + (-5.11 - 5.11i)T + 31iT^{2} \)
37 \( 1 + (4.95 - 4.95i)T - 37iT^{2} \)
41 \( 1 + (1.70 - 1.70i)T - 41iT^{2} \)
43 \( 1 + 6.12T + 43T^{2} \)
47 \( 1 + 0.483iT - 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 - 3.16iT - 59T^{2} \)
61 \( 1 + (-1.38 + 1.38i)T - 61iT^{2} \)
67 \( 1 + 1.11iT - 67T^{2} \)
71 \( 1 + (-4.05 - 4.05i)T + 71iT^{2} \)
73 \( 1 + (0.756 - 0.756i)T - 73iT^{2} \)
79 \( 1 + (5.54 - 5.54i)T - 79iT^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + (-10.3 + 10.3i)T - 97iT^{2} \)
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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

−8.867841931444168000563482213798, −8.151333461796102813554912783473, −7.06453906092609453786606408776, −6.56833283848783454995502714717, −6.01090258192854083125871567266, −4.91832658166897872272831963521, −4.07648669331832906789227848674, −3.31351647083475372158889656722, −2.11463148521374778747916152682, −1.13942053133217804523679462614, 0.52941497668344670358018141093, 2.16712330002430815024563793713, 2.89810705007453584583553046695, 3.49058415699955192328249301953, 5.11970425654708574935894550857, 5.67673078642431250237961802515, 6.06509168309044453518193968387, 6.99795238660176360724521598946, 7.912363240840623859488579573504, 8.713401770992027863414243975383

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