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Properties

Label 2-3040-3040.189-c0-0-2
Degree $2$
Conductor $3040$
Sign $-0.471 - 0.881i$
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  + (−0.956 + 0.290i)2-s + (0.871 + 0.360i)3-s + (0.831 − 0.555i)4-s + (0.382 + 0.923i)5-s + (−0.938 − 0.0924i)6-s + (−0.634 + 0.773i)8-s + (−0.0785 − 0.0785i)9-s + (−0.634 − 0.773i)10-s + (0.360 − 0.149i)11-s + (0.924 − 0.183i)12-s + (−0.761 + 1.83i)13-s + 0.942i·15-s + (0.382 − 0.923i)16-s + (0.0980 + 0.0523i)18-s + (−0.382 + 0.923i)19-s + (0.831 + 0.555i)20-s + ⋯
L(s)  = 1  + (−0.956 + 0.290i)2-s + (0.871 + 0.360i)3-s + (0.831 − 0.555i)4-s + (0.382 + 0.923i)5-s + (−0.938 − 0.0924i)6-s + (−0.634 + 0.773i)8-s + (−0.0785 − 0.0785i)9-s + (−0.634 − 0.773i)10-s + (0.360 − 0.149i)11-s + (0.924 − 0.183i)12-s + (−0.761 + 1.83i)13-s + 0.942i·15-s + (0.382 − 0.923i)16-s + (0.0980 + 0.0523i)18-s + (−0.382 + 0.923i)19-s + (0.831 + 0.555i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-0.471 - 0.881i$
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: \((2,\ 3040,\ (\ :0),\ -0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010386763\)
\(L(\frac12)\) \(\approx\) \(1.010386763\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.956 - 0.290i)T \)
5 \( 1 + (-0.382 - 0.923i)T \)
19 \( 1 + (0.382 - 0.923i)T \)
good3 \( 1 + (-0.871 - 0.360i)T + (0.707 + 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-0.360 + 0.149i)T + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.761 - 1.83i)T + (-0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.222 - 0.536i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.62 - 0.674i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-1.81 - 0.750i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (-1.42 - 0.591i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 - 0.196T + T^{2} \)
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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

−9.163980795859505787574810193008, −8.529182075997102467757902797723, −7.77125298373777711846309464390, −6.87692482626891387280822057976, −6.49035220494985368783869437941, −5.61331862493034753561104382441, −4.30908113610089628417178114359, −3.38808933503803121215862376225, −2.43291956416617847574705458521, −1.77073380192371602501402105405, 0.72568875372065699814535023896, 1.97702351905523736038437915001, 2.65964146097347154112870946976, 3.52509084678883565088301187815, 4.82569316982517014941965042666, 5.63982490462760850829577538526, 6.64906229916383202686920594385, 7.58167513855208584196958301240, 8.066726660139136723864060245774, 8.593463689758460033649765500812

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