Properties

Label 2-1620-15.14-c2-0-37
Degree $2$
Conductor $1620$
Sign $0.195 + 0.980i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.90 − 0.978i)5-s + 7.34i·7-s − 9.64i·11-s − 16.6i·13-s + 15.4·17-s − 17.4·19-s − 10.5·23-s + (23.0 − 9.59i)25-s − 41.8i·29-s − 48.7·31-s + (7.19 + 36.0i)35-s − 20.4i·37-s − 27.9i·41-s − 30.2i·43-s + 42.1·47-s + ⋯
L(s)  = 1  + (0.980 − 0.195i)5-s + 1.04i·7-s − 0.876i·11-s − 1.28i·13-s + 0.910·17-s − 0.920·19-s − 0.460·23-s + (0.923 − 0.383i)25-s − 1.44i·29-s − 1.57·31-s + (0.205 + 1.02i)35-s − 0.553i·37-s − 0.680i·41-s − 0.702i·43-s + 0.896·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.195 + 0.980i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.989093318\)
\(L(\frac12)\) \(\approx\) \(1.989093318\)
\(L(2)\) 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 \( 1 \)
5 \( 1 + (-4.90 + 0.978i)T \)
good7 \( 1 - 7.34iT - 49T^{2} \)
11 \( 1 + 9.64iT - 121T^{2} \)
13 \( 1 + 16.6iT - 169T^{2} \)
17 \( 1 - 15.4T + 289T^{2} \)
19 \( 1 + 17.4T + 361T^{2} \)
23 \( 1 + 10.5T + 529T^{2} \)
29 \( 1 + 41.8iT - 841T^{2} \)
31 \( 1 + 48.7T + 961T^{2} \)
37 \( 1 + 20.4iT - 1.36e3T^{2} \)
41 \( 1 + 27.9iT - 1.68e3T^{2} \)
43 \( 1 + 30.2iT - 1.84e3T^{2} \)
47 \( 1 - 42.1T + 2.20e3T^{2} \)
53 \( 1 + 90.4T + 2.80e3T^{2} \)
59 \( 1 - 55.4iT - 3.48e3T^{2} \)
61 \( 1 - 116.T + 3.72e3T^{2} \)
67 \( 1 + 17.2iT - 4.48e3T^{2} \)
71 \( 1 + 38.4iT - 5.04e3T^{2} \)
73 \( 1 + 7.38iT - 5.32e3T^{2} \)
79 \( 1 - 23.6T + 6.24e3T^{2} \)
83 \( 1 + 4.26T + 6.88e3T^{2} \)
89 \( 1 - 145. iT - 7.92e3T^{2} \)
97 \( 1 + 102. iT - 9.40e3T^{2} \)
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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

−8.982164049197307012129893356532, −8.400139913001936736968844607132, −7.58111699516862020037610582211, −6.28804705352548780309752549697, −5.66455654437920638272786867444, −5.33889524019758153469510265168, −3.85648447165052706967879989335, −2.78305091354442771059311572220, −1.97025527191099256873068314550, −0.52336286120567651062817104285, 1.34378042707796067782821841208, 2.11642586685660186452331291336, 3.47550713161477535920428677442, 4.39797091754734484062896092184, 5.22436997461431864853083393106, 6.32758279645447452573041668668, 6.92925395793980168783316261130, 7.59462023322376793133180091533, 8.741102047523982926230908809116, 9.531171528459028898051151012507

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