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Properties

Label 2-2940-35.13-c1-0-27
Degree $2$
Conductor $2940$
Sign $0.952 + 0.303i$
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  + (0.707 − 0.707i)3-s + (2.07 + 0.836i)5-s − 1.00i·9-s + 2.28·11-s + (0.277 − 0.277i)13-s + (2.05 − 0.874i)15-s + (0.0556 + 0.0556i)17-s + 3.08·19-s + (−0.206 − 0.206i)23-s + (3.60 + 3.46i)25-s + (−0.707 − 0.707i)27-s − 3.92i·29-s − 1.24i·31-s + (1.61 − 1.61i)33-s + (1.47 − 1.47i)37-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.927 + 0.374i)5-s − 0.333i·9-s + 0.690·11-s + (0.0769 − 0.0769i)13-s + (0.531 − 0.225i)15-s + (0.0134 + 0.0134i)17-s + 0.707·19-s + (−0.0431 − 0.0431i)23-s + (0.720 + 0.693i)25-s + (−0.136 − 0.136i)27-s − 0.729i·29-s − 0.223i·31-s + (0.281 − 0.281i)33-s + (0.242 − 0.242i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.952 + 0.303i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 0.952 + 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.812985221\)
\(L(\frac12)\) \(\approx\) \(2.812985221\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-2.07 - 0.836i)T \)
7 \( 1 \)
good11 \( 1 - 2.28T + 11T^{2} \)
13 \( 1 + (-0.277 + 0.277i)T - 13iT^{2} \)
17 \( 1 + (-0.0556 - 0.0556i)T + 17iT^{2} \)
19 \( 1 - 3.08T + 19T^{2} \)
23 \( 1 + (0.206 + 0.206i)T + 23iT^{2} \)
29 \( 1 + 3.92iT - 29T^{2} \)
31 \( 1 + 1.24iT - 31T^{2} \)
37 \( 1 + (-1.47 + 1.47i)T - 37iT^{2} \)
41 \( 1 + 0.184iT - 41T^{2} \)
43 \( 1 + (0.541 + 0.541i)T + 43iT^{2} \)
47 \( 1 + (-5.46 - 5.46i)T + 47iT^{2} \)
53 \( 1 + (-7.91 - 7.91i)T + 53iT^{2} \)
59 \( 1 + 5.78T + 59T^{2} \)
61 \( 1 + 6.85iT - 61T^{2} \)
67 \( 1 + (7.59 - 7.59i)T - 67iT^{2} \)
71 \( 1 + 3.95T + 71T^{2} \)
73 \( 1 + (-5.30 + 5.30i)T - 73iT^{2} \)
79 \( 1 + 13.0iT - 79T^{2} \)
83 \( 1 + (-7.50 + 7.50i)T - 83iT^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + (-6.10 - 6.10i)T + 97iT^{2} \)
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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

−8.998425435717462997038365557870, −7.80290264617718858884880332207, −7.29257361756238271322371967940, −6.31554209789422689869306505400, −5.94089945192418550048373110371, −4.88094617331424727433009013316, −3.82116328566611762179353904521, −2.90672059729920968239729081654, −2.05689628564100966521099128865, −1.04656096698032889962806871952, 1.12190176306293664297943947648, 2.12521108016394728500055658791, 3.14595146037933047603525225830, 4.04888139752899077089285876100, 4.97203143334087704356050050031, 5.61678374007065040744030932949, 6.50007038668721317713347064588, 7.24627344413222278436371838101, 8.249861596831485326169297105749, 8.945204709875001793716986338111

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