Properties

Label 2-2940-2940.1319-c0-0-1
Degree $2$
Conductor $2940$
Sign $0.672 - 0.740i$
Analytic cond. $1.46725$
Root an. cond. $1.21130$
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  + (−0.563 + 0.826i)2-s + (−0.866 + 0.5i)3-s + (−0.365 − 0.930i)4-s + (0.955 − 0.294i)5-s + (0.0747 − 0.997i)6-s + (0.930 − 0.365i)7-s + (0.974 + 0.222i)8-s + (0.499 − 0.866i)9-s + (−0.294 + 0.955i)10-s + (0.781 + 0.623i)12-s + (−0.222 + 0.974i)14-s + (−0.680 + 0.733i)15-s + (−0.733 + 0.680i)16-s + (0.433 + 0.900i)18-s + (−0.623 − 0.781i)20-s + (−0.623 + 0.781i)21-s + ⋯
L(s)  = 1  + (−0.563 + 0.826i)2-s + (−0.866 + 0.5i)3-s + (−0.365 − 0.930i)4-s + (0.955 − 0.294i)5-s + (0.0747 − 0.997i)6-s + (0.930 − 0.365i)7-s + (0.974 + 0.222i)8-s + (0.499 − 0.866i)9-s + (−0.294 + 0.955i)10-s + (0.781 + 0.623i)12-s + (−0.222 + 0.974i)14-s + (−0.680 + 0.733i)15-s + (−0.733 + 0.680i)16-s + (0.433 + 0.900i)18-s + (−0.623 − 0.781i)20-s + (−0.623 + 0.781i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.672 - 0.740i$
Analytic conductor: \(1.46725\)
Root analytic conductor: \(1.21130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :0),\ 0.672 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9188529059\)
\(L(\frac12)\) \(\approx\) \(0.9188529059\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.563 - 0.826i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.955 + 0.294i)T \)
7 \( 1 + (-0.930 + 0.365i)T \)
good11 \( 1 + (-0.988 - 0.149i)T^{2} \)
13 \( 1 + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.218 - 1.44i)T + (-0.955 - 0.294i)T^{2} \)
29 \( 1 + (0.233 - 0.185i)T + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.733 + 0.680i)T^{2} \)
41 \( 1 + (-0.0332 + 0.145i)T + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.443 + 1.94i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-1.61 - 1.09i)T + (0.365 + 0.930i)T^{2} \)
53 \( 1 + (-0.733 + 0.680i)T^{2} \)
59 \( 1 + (-0.826 - 0.563i)T^{2} \)
61 \( 1 + (-1.26 - 0.496i)T + (0.733 + 0.680i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.365 - 0.930i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.67 + 0.807i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.147 - 1.97i)T + (-0.988 + 0.149i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.118020543351695627000337672775, −8.346718787872894355575246205021, −7.38359189791540344044870754535, −6.80804972810409833804052002721, −5.75160779315229460275070202917, −5.47598122170647210725452853912, −4.71551115384355512576232884673, −3.85579565868876917705823738971, −1.96803902131978864238773825645, −1.03257249390713415999392981758, 1.09674493272970446665423129830, 2.05121449815769782129690457136, 2.66994714230595808102582353391, 4.19425686046220781605018826537, 4.97737904810178113030656454550, 5.75154831332503104503028446005, 6.63387124954753683647305180115, 7.36342245435729219099281085236, 8.249330061611607744287084246539, 8.795418123629872161419124918178

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