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Properties

Label 2-2940-35.4-c1-0-24
Degree $2$
Conductor $2940$
Sign $-0.174 + 0.984i$
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  + (−0.866 − 0.5i)3-s + (−0.490 − 2.18i)5-s + (0.499 + 0.866i)9-s + (−1.90 + 3.29i)11-s + 2.31i·13-s + (−0.666 + 2.13i)15-s + (0.324 + 0.187i)17-s + (−0.453 − 0.785i)19-s + (1.13 − 0.656i)23-s + (−4.51 + 2.13i)25-s − 0.999i·27-s + 6.15·29-s + (2.81 − 4.87i)31-s + (3.29 − 1.90i)33-s + (4.96 − 2.86i)37-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (−0.219 − 0.975i)5-s + (0.166 + 0.288i)9-s + (−0.572 + 0.992i)11-s + 0.641i·13-s + (−0.171 + 0.551i)15-s + (0.0787 + 0.0454i)17-s + (−0.103 − 0.180i)19-s + (0.236 − 0.136i)23-s + (−0.903 + 0.427i)25-s − 0.192i·27-s + 1.14·29-s + (0.505 − 0.874i)31-s + (0.572 − 0.330i)33-s + (0.816 − 0.471i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -0.174 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.069779588\)
\(L(\frac12)\) \(\approx\) \(1.069779588\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.490 + 2.18i)T \)
7 \( 1 \)
good11 \( 1 + (1.90 - 3.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.31iT - 13T^{2} \)
17 \( 1 + (-0.324 - 0.187i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.453 + 0.785i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.13 + 0.656i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.15T + 29T^{2} \)
31 \( 1 + (-2.81 + 4.87i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.96 + 2.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.199T + 41T^{2} \)
43 \( 1 + 12.4iT - 43T^{2} \)
47 \( 1 + (-9.35 + 5.40i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.65 - 1.53i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.14 - 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.08 - 1.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.81 + 2.78i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.88T + 71T^{2} \)
73 \( 1 + (5.54 + 3.20i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.61 + 13.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.99iT - 83T^{2} \)
89 \( 1 + (-2.50 - 4.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.36iT - 97T^{2} \)
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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

−8.650862685323004735626233545059, −7.60009405592675688526980278257, −7.23056881234933800461513443153, −6.16714379143918252563431761775, −5.44934063286896498598252864273, −4.56990183506014722093628878528, −4.16834506080918915586743211746, −2.61817874656021738678507710362, −1.65348881896535124991537219508, −0.44267022207998622192508554521, 0.997242369753798362786843221450, 2.73770427535286187089785862826, 3.18052993371117815741993705390, 4.27329926246189822820258349720, 5.16112409413007078349935471030, 6.05615752109812701164909843257, 6.48043983667034417260760033948, 7.52369514196073176740852692751, 8.091127118227715759016659781543, 8.911205550806853808556313899647

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