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Properties

Label 2-2940-7.2-c1-0-22
Degree $2$
Conductor $2940$
Sign $-0.991 - 0.126i$
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 + (1 + 1.73i)11-s − 13-s − 0.999·15-s + (−2 − 3.46i)17-s + (−0.5 + 0.866i)19-s + (−2 + 3.46i)23-s + (−0.499 − 0.866i)25-s + 0.999·27-s + (−2.5 − 4.33i)31-s + (0.999 − 1.73i)33-s + (2.5 − 4.33i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s − 0.277·13-s − 0.258·15-s + (−0.485 − 0.840i)17-s + (−0.114 + 0.198i)19-s + (−0.417 + 0.722i)23-s + (−0.0999 − 0.173i)25-s + 0.192·27-s + (−0.449 − 0.777i)31-s + (0.174 − 0.301i)33-s + (0.410 − 0.711i)37-s + (0.0800 + 0.138i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3887476780\)
\(L(\frac12)\) \(\approx\) \(0.3887476780\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 + 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 18T + 83T^{2} \)
89 \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.226756134009800347984633034079, −7.59848385375492925788663742825, −6.83827257610993932247739533373, −6.13618844007862103860619325332, −5.25395248528846265708673647081, −4.60774761753197938999649507764, −3.56614950918298652156823052281, −2.34294410507449610390057520373, −1.53030772063639054800415935490, −0.12255910250252248636324060930, 1.53349977881221406778397546784, 2.74198201229855707017070903529, 3.61161943977434104734857719215, 4.48363188894129312230115167641, 5.26774778465545613388498443826, 6.28366609631346763648242563772, 6.56355347164477265007304470630, 7.68978854458079699461056854830, 8.505776427995543044288445542869, 9.133907268215894273559620794667

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