Properties

Label 2-2940-2940.1559-c0-0-1
Degree $2$
Conductor $2940$
Sign $0.918 - 0.394i$
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  + (−0.680 + 0.733i)2-s + (−0.974 + 0.222i)3-s + (−0.0747 − 0.997i)4-s + (−0.365 − 0.930i)5-s + (0.5 − 0.866i)6-s + (0.997 − 0.0747i)7-s + (0.781 + 0.623i)8-s + (0.900 − 0.433i)9-s + (0.930 + 0.365i)10-s + (0.294 + 0.955i)12-s + (−0.623 + 0.781i)14-s + (0.563 + 0.826i)15-s + (−0.988 + 0.149i)16-s + (−0.294 + 0.955i)18-s + (−0.900 + 0.433i)20-s + (−0.955 + 0.294i)21-s + ⋯
L(s)  = 1  + (−0.680 + 0.733i)2-s + (−0.974 + 0.222i)3-s + (−0.0747 − 0.997i)4-s + (−0.365 − 0.930i)5-s + (0.5 − 0.866i)6-s + (0.997 − 0.0747i)7-s + (0.781 + 0.623i)8-s + (0.900 − 0.433i)9-s + (0.930 + 0.365i)10-s + (0.294 + 0.955i)12-s + (−0.623 + 0.781i)14-s + (0.563 + 0.826i)15-s + (−0.988 + 0.149i)16-s + (−0.294 + 0.955i)18-s + (−0.900 + 0.433i)20-s + (−0.955 + 0.294i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.918 - 0.394i$
Analytic conductor: \(1.46725\)
Root analytic conductor: \(1.21130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :0),\ 0.918 - 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6718785053\)
\(L(\frac12)\) \(\approx\) \(0.6718785053\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.680 - 0.733i)T \)
3 \( 1 + (0.974 - 0.222i)T \)
5 \( 1 + (0.365 + 0.930i)T \)
7 \( 1 + (-0.997 + 0.0747i)T \)
good11 \( 1 + (0.826 - 0.563i)T^{2} \)
13 \( 1 + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.365 - 0.930i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.11 - 1.63i)T + (-0.365 + 0.930i)T^{2} \)
29 \( 1 + (-0.488 - 1.01i)T + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (-1.19 + 1.49i)T + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.367 - 0.460i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (1.14 + 1.06i)T + (0.0747 + 0.997i)T^{2} \)
53 \( 1 + (-0.988 + 0.149i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (-0.297 - 0.0222i)T + (0.988 + 0.149i)T^{2} \)
67 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.0747 - 0.997i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.443 - 1.94i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-1.57 - 0.487i)T + (0.826 + 0.563i)T^{2} \)
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

−9.062034930351357562228463174763, −8.146160561294327504940410864286, −7.49528103699855243204912619618, −6.87511703989865673437503932703, −5.78623695027667352102175315045, −5.17100040864429622178110911045, −4.77383249792225607069871963775, −3.76903290330163612035432888413, −1.73003842478865884293475793944, −0.917178553529880116009620694004, 0.905919265284063602320130955494, 2.13899572789584376804263885506, 2.97989565350601833832746402777, 4.31625106348135851261748404210, 4.69516890277244055016382657574, 6.03892954811838571544121189788, 6.74044257129270952810283903431, 7.53578077685794450856560059974, 8.019899560363266684070669087695, 8.879774712446418361758632275422

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