Properties

Label 2-3040-3040.189-c0-0-3
Degree $2$
Conductor $3040$
Sign $0.881 - 0.471i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
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  + (−0.290 − 0.956i)2-s + (1.62 + 0.674i)3-s + (−0.831 + 0.555i)4-s + (0.382 + 0.923i)5-s + (0.172 − 1.75i)6-s + (0.773 + 0.634i)8-s + (1.49 + 1.49i)9-s + (0.773 − 0.634i)10-s + (−0.360 + 0.149i)11-s + (−1.72 + 0.344i)12-s + (0.0750 − 0.181i)13-s + 1.76i·15-s + (0.382 − 0.923i)16-s + (0.995 − 1.86i)18-s + (−0.382 + 0.923i)19-s + (−0.831 − 0.555i)20-s + ⋯
L(s)  = 1  + (−0.290 − 0.956i)2-s + (1.62 + 0.674i)3-s + (−0.831 + 0.555i)4-s + (0.382 + 0.923i)5-s + (0.172 − 1.75i)6-s + (0.773 + 0.634i)8-s + (1.49 + 1.49i)9-s + (0.773 − 0.634i)10-s + (−0.360 + 0.149i)11-s + (−1.72 + 0.344i)12-s + (0.0750 − 0.181i)13-s + 1.76i·15-s + (0.382 − 0.923i)16-s + (0.995 − 1.86i)18-s + (−0.382 + 0.923i)19-s + (−0.831 − 0.555i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.820869121\)
\(L(\frac12)\) \(\approx\) \(1.820869121\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.290 + 0.956i)T \)
5 \( 1 + (-0.382 - 0.923i)T \)
19 \( 1 + (0.382 - 0.923i)T \)
good3 \( 1 + (-1.62 - 0.674i)T + (0.707 + 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.360 - 0.149i)T + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (-0.0750 + 0.181i)T + (-0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.732 + 1.76i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.871 + 0.360i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (1.81 + 0.750i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (-1.17 - 0.485i)T + (0.707 + 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - 1.99T + T^{2} \)
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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

−9.082428053786967065058042889382, −8.430647904522182213601122536004, −7.74518877350029399848694700527, −7.12748665856193680598643545099, −5.75406872092554815868556879245, −4.71051862496900823953715226800, −3.71039994537801425543485290448, −3.38337080683637280565003336870, −2.37127930909659589602375687616, −1.91233316741232739971932007705, 1.09571823621088083909084422720, 2.06482820222978530600961232488, 3.16918798840704130802337786731, 4.28089899939288551788214861373, 4.96411645796368208347390514336, 6.04986552894806377537852884561, 6.78138011120974946961421646807, 7.54977998077646741234007943939, 8.161420321525966629772795943339, 8.744310127530641835974211643476

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