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Properties

Label 2-2940-420.59-c0-0-17
Degree $2$
Conductor $2940$
Sign $0.350 + 0.936i$
Analytic cond. $1.46725$
Root an. cond. $1.21130$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.991 − 0.130i)2-s + (0.5 − 0.866i)3-s + (0.965 − 0.258i)4-s + (−0.866 + 0.5i)5-s + (0.382 − 0.923i)6-s + (0.923 − 0.382i)8-s + (−0.499 − 0.866i)9-s + (−0.793 + 0.608i)10-s + (0.258 − 0.965i)12-s + 0.999i·15-s + (0.866 − 0.5i)16-s + (1.22 + 0.707i)17-s + (−0.608 − 0.793i)18-s + (−0.923 − 1.60i)19-s + (−0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.991 − 0.130i)2-s + (0.5 − 0.866i)3-s + (0.965 − 0.258i)4-s + (−0.866 + 0.5i)5-s + (0.382 − 0.923i)6-s + (0.923 − 0.382i)8-s + (−0.499 − 0.866i)9-s + (−0.793 + 0.608i)10-s + (0.258 − 0.965i)12-s + 0.999i·15-s + (0.866 − 0.5i)16-s + (1.22 + 0.707i)17-s + (−0.608 − 0.793i)18-s + (−0.923 − 1.60i)19-s + (−0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.350 + 0.936i$
Analytic conductor: \(1.46725\)
Root analytic conductor: \(1.21130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (2579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :0),\ 0.350 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.426422176\)
\(L(\frac12)\) \(\approx\) \(2.426422176\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.991 + 0.130i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 \)
good11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.549733625397123202767247442948, −7.73118011759783599237644311511, −7.29366616578235789485980002594, −6.48075805577912380773011346439, −5.94537812736361547352667501870, −4.70853535649119924820394863076, −3.97313375369669587171254352567, −3.01369118845794401282211949614, −2.53488597608419041218177883529, −1.14446927074502573614271649256, 1.70291093884785756448904253722, 3.07456742464821395343736080153, 3.57410655492189257448500510195, 4.34249723392892386490136891792, 5.08723168298194931572480092905, 5.67057755231503233268117774839, 6.80030518614287655166835171678, 7.77738149435025076458447621033, 8.096371762498164160320861364663, 8.960883788105082045486427845344

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