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Properties

Label 2-2940-7.2-c1-0-5
Degree $2$
Conductor $2940$
Sign $0.749 - 0.661i$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
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  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)11-s + 2.82·13-s − 0.999·15-s + (3.41 + 5.91i)17-s + (−1.29 + 2.23i)19-s + (2.29 − 3.97i)23-s + (−0.499 − 0.866i)25-s + 0.999·27-s − 7.65·29-s + (2.12 + 3.67i)31-s + (0.999 − 1.73i)33-s + (−3.24 + 5.61i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s + 0.784·13-s − 0.258·15-s + (0.828 + 1.43i)17-s + (−0.296 + 0.513i)19-s + (0.478 − 0.828i)23-s + (−0.0999 − 0.173i)25-s + 0.192·27-s − 1.42·29-s + (0.381 + 0.659i)31-s + (0.174 − 0.301i)33-s + (−0.533 + 0.923i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.749 - 0.661i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 0.749 - 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.542425692\)
\(L(\frac12)\) \(\approx\) \(1.542425692\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + (-3.41 - 5.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.29 - 2.23i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.29 + 3.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 + (-2.12 - 3.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.24 - 5.61i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 9.65T + 43T^{2} \)
47 \( 1 + (3.24 - 5.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.70 - 6.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.41 - 2.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.94 - 8.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.585 - 1.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.48T + 71T^{2} \)
73 \( 1 + (-6.24 - 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.828T + 83T^{2} \)
89 \( 1 + (0.414 - 0.717i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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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

−8.583466607157045830012872064829, −8.253232801315106600292989329148, −7.29100149077528568997697598064, −6.45890279959858508310891655156, −5.91393360062782098841785524896, −5.07462078464143423098010044014, −4.12920685650424859562075424137, −3.25661455939893867723139764787, −1.86923119065494120340392578603, −1.21916701030272720931253362066, 0.54965636173086502522842027987, 1.96362349349949316814313105750, 3.28099395583183184095531491536, 3.67332716597362280027911475818, 4.98255630089778293285607006109, 5.48554150416850078960644846878, 6.36893608758662363357931638330, 7.06521828040523035023526959005, 7.904451383827140118217318181443, 8.833005511559376450380795537926

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