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Properties

Label 2-2940-420.299-c0-0-0
Degree $2$
Conductor $2940$
Sign $-0.126 + 0.991i$
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  + (−0.130 + 0.991i)2-s + (0.5 + 0.866i)3-s + (−0.965 − 0.258i)4-s + (−0.866 − 0.5i)5-s + (−0.923 + 0.382i)6-s + (0.382 − 0.923i)8-s + (−0.499 + 0.866i)9-s + (0.608 − 0.793i)10-s + (−0.258 − 0.965i)12-s − 0.999i·15-s + (0.866 + 0.5i)16-s + (−1.22 + 0.707i)17-s + (−0.793 − 0.608i)18-s + (−0.382 + 0.662i)19-s + (0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (0.5 + 0.866i)3-s + (−0.965 − 0.258i)4-s + (−0.866 − 0.5i)5-s + (−0.923 + 0.382i)6-s + (0.382 − 0.923i)8-s + (−0.499 + 0.866i)9-s + (0.608 − 0.793i)10-s + (−0.258 − 0.965i)12-s − 0.999i·15-s + (0.866 + 0.5i)16-s + (−1.22 + 0.707i)17-s + (−0.793 − 0.608i)18-s + (−0.382 + 0.662i)19-s + (0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.126 + 0.991i$
Analytic conductor: \(1.46725\)
Root analytic conductor: \(1.21130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :0),\ -0.126 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1578322051\)
\(L(\frac12)\) \(\approx\) \(0.1578322051\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.130 - 0.991i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 \)
good11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.662 + 0.382i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−9.299355830832147260023270955152, −8.473132157746183186251436191869, −8.171305101146448568546956083319, −7.50376727013487641128898040604, −6.38119729853053710556050495218, −5.71377028118070103123528251627, −4.62036743062570953049009707706, −4.22382272904591002310996570900, −3.56813833295127244360787984638, −2.02598639986892776339342491866, 0.092551901307648169805089912359, 1.68536399782768725762792688777, 2.56096098130036177506272042068, 3.37024751109406227820345813409, 4.09362584519053887776093259266, 5.06747445393391765136930047598, 6.28875232773598617524193857940, 7.13539602942402355519234143939, 7.68858843419654413919627756515, 8.548247374732503199712560331641

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