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Properties

Label 2-2940-35.4-c1-0-37
Degree $2$
Conductor $2940$
Sign $-0.999 - 0.00125i$
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 + (2.06 − 0.861i)5-s + (0.499 + 0.866i)9-s + (−1.46 + 2.53i)11-s − 5.76i·13-s + (−2.21 − 0.285i)15-s + (0.205 + 0.118i)17-s + (−2.37 − 4.10i)19-s + (−5.86 + 3.38i)23-s + (3.51 − 3.55i)25-s − 0.999i·27-s − 6.03·29-s + (−5.26 + 9.12i)31-s + (2.53 − 1.46i)33-s + (−2.66 + 1.53i)37-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.922 − 0.385i)5-s + (0.166 + 0.288i)9-s + (−0.442 + 0.765i)11-s − 1.60i·13-s + (−0.572 − 0.0737i)15-s + (0.0497 + 0.0287i)17-s + (−0.543 − 0.941i)19-s + (−1.22 + 0.705i)23-s + (0.703 − 0.711i)25-s − 0.192i·27-s − 1.12·29-s + (−0.946 + 1.63i)31-s + (0.442 − 0.255i)33-s + (−0.437 + 0.252i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.999 - 0.00125i$
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: \((2,\ 2940,\ (\ :1/2),\ -0.999 - 0.00125i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4622668602\)
\(L(\frac12)\) \(\approx\) \(0.4622668602\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.06 + 0.861i)T \)
7 \( 1 \)
good11 \( 1 + (1.46 - 2.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.76iT - 13T^{2} \)
17 \( 1 + (-0.205 - 0.118i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.37 + 4.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.86 - 3.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.03T + 29T^{2} \)
31 \( 1 + (5.26 - 9.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.66 - 1.53i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.06T + 41T^{2} \)
43 \( 1 + 9.79iT - 43T^{2} \)
47 \( 1 + (3.21 - 1.85i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.53 - 5.50i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.65 + 8.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.06 + 7.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.94 - 1.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + (3.28 + 1.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.00873 - 0.0151i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.30iT - 83T^{2} \)
89 \( 1 + (8.97 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.31iT - 97T^{2} \)
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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

−8.401138100230121820793886350727, −7.50503940893680251394517791705, −6.90764220423007946660847248943, −5.86721896393339285264526526739, −5.40441509018860193748310665278, −4.77171121445860081367816298609, −3.51984447657481856489064300210, −2.37601644291226776913064027508, −1.54773643548128754606562638391, −0.14306321506209171624113913293, 1.65224983227590508885924082054, 2.42651213997932321073525648924, 3.74127574002312422549846561399, 4.37463324429047903670197240299, 5.60316249241580619226028790281, 5.94049124817718357603676482987, 6.67490092578397626150334288603, 7.52593575060309441580358327281, 8.513704020427334635144522954169, 9.233527310359029687451331332143

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