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Properties

Label 2-2940-7.4-c1-0-12
Degree $2$
Conductor $2940$
Sign $0.968 + 0.250i$
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  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (−3 + 5.19i)11-s + 4·13-s − 0.999·15-s + (3 − 5.19i)17-s + (1 + 1.73i)19-s + (−0.499 + 0.866i)25-s − 0.999·27-s + 6·29-s + (−5 + 8.66i)31-s + (3 + 5.19i)33-s + (−1 − 1.73i)37-s + (2 − 3.46i)39-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.166 − 0.288i)9-s + (−0.904 + 1.56i)11-s + 1.10·13-s − 0.258·15-s + (0.727 − 1.26i)17-s + (0.229 + 0.397i)19-s + (−0.0999 + 0.173i)25-s − 0.192·27-s + 1.11·29-s + (−0.898 + 1.55i)31-s + (0.522 + 0.904i)33-s + (−0.164 − 0.284i)37-s + (0.320 − 0.554i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.934750632\)
\(L(\frac12)\) \(\approx\) \(1.934750632\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16T + 97T^{2} \)
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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

−8.588018757815467786693534201791, −7.963704996842236722158459408661, −7.23681401287380703088312547479, −6.71569455644001443332986457539, −5.47501285781594510492661481540, −4.98615329059275740491042969478, −3.92606990802085044550126518183, −2.98995153933347679428242606842, −1.99068776389576374274500549912, −0.917705764820059574521927978956, 0.796694075166793380005012426771, 2.33236940834690638887703922909, 3.38778091283191629358938021367, 3.71655493578559020052016504233, 4.90275003282740461299421069852, 5.91558172568666006639230915668, 6.18771835736924054192121216631, 7.52593154822057604177830306081, 8.167712440366459125359335610136, 8.613748919606232809341210087440

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