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Properties

Label 2-2960-740.427-c0-0-0
Degree $2$
Conductor $2960$
Sign $0.171 + 0.985i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 − 0.642i)5-s + (0.342 − 0.939i)9-s + (−0.342 − 0.939i)13-s + (−1.20 − 0.439i)17-s + (0.173 − 0.984i)25-s + (0.816 + 0.218i)29-s + (−0.939 + 0.342i)37-s + (−0.439 − 1.20i)41-s + (−0.342 − 0.939i)45-s + (−0.984 + 0.173i)49-s + (−0.0451 + 0.515i)53-s + (1.64 + 0.766i)61-s + (−0.866 − 0.5i)65-s + (1.36 + 1.36i)73-s + (−0.766 − 0.642i)81-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)5-s + (0.342 − 0.939i)9-s + (−0.342 − 0.939i)13-s + (−1.20 − 0.439i)17-s + (0.173 − 0.984i)25-s + (0.816 + 0.218i)29-s + (−0.939 + 0.342i)37-s + (−0.439 − 1.20i)41-s + (−0.342 − 0.939i)45-s + (−0.984 + 0.173i)49-s + (−0.0451 + 0.515i)53-s + (1.64 + 0.766i)61-s + (−0.866 − 0.5i)65-s + (1.36 + 1.36i)73-s + (−0.766 − 0.642i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.171 + 0.985i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.171 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.311484965\)
\(L(\frac12)\) \(\approx\) \(1.311484965\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (0.939 - 0.342i)T \)
good3 \( 1 + (-0.342 + 0.939i)T^{2} \)
7 \( 1 + (0.984 - 0.173i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.342 + 0.939i)T + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (1.20 + 0.439i)T + (0.766 + 0.642i)T^{2} \)
19 \( 1 + (-0.342 + 0.939i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.816 - 0.218i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.0451 - 0.515i)T + (-0.984 - 0.173i)T^{2} \)
59 \( 1 + (-0.984 - 0.173i)T^{2} \)
61 \( 1 + (-1.64 - 0.766i)T + (0.642 + 0.766i)T^{2} \)
67 \( 1 + (-0.984 + 0.173i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
79 \( 1 + (0.984 - 0.173i)T^{2} \)
83 \( 1 + (-0.642 + 0.766i)T^{2} \)
89 \( 1 + (0.173 - 1.98i)T + (-0.984 - 0.173i)T^{2} \)
97 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.783661860868405901258066792857, −8.232552547904586294402023349777, −7.04944628721077734589331546444, −6.56837461940898213712567600285, −5.62591886743066498114785810712, −4.97208602100862139410499988551, −4.11479688792705774909607988400, −3.04264022763408933303666116816, −2.04561414614780268845462574423, −0.806386495442069763455938371966, 1.80077700636066922378148926124, 2.29932715202899270116373944516, 3.46518937587832876619450212234, 4.58351529950196284383368648538, 5.12527658526776176655511040584, 6.31218291164998781695117588875, 6.68629391259267889602627522822, 7.50178154754754792409883261711, 8.385335083096028687311387146519, 9.124327376005589681981559444880

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