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Properties

Label 2-1620-9.4-c3-0-13
Degree $2$
Conductor $1620$
Sign $0.766 - 0.642i$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.5 + 4.33i)5-s + (−0.474 − 0.822i)7-s + (−31.3 − 54.2i)11-s + (−3.47 + 6.01i)13-s − 103.·17-s + 73.8·19-s + (−43.0 + 74.6i)23-s + (−12.5 − 21.6i)25-s + (27.7 + 48.1i)29-s + (−99.9 + 173. i)31-s + 4.74·35-s − 18.9·37-s + (28.6 − 49.6i)41-s + (−148. − 256. i)43-s + (−28.7 − 49.8i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (−0.0256 − 0.0444i)7-s + (−0.858 − 1.48i)11-s + (−0.0741 + 0.128i)13-s − 1.47·17-s + 0.891·19-s + (−0.390 + 0.676i)23-s + (−0.100 − 0.173i)25-s + (0.177 + 0.308i)29-s + (−0.578 + 1.00i)31-s + 0.0229·35-s − 0.0840·37-s + (0.109 − 0.189i)41-s + (−0.524 − 0.909i)43-s + (−0.0892 − 0.154i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.282681243\)
\(L(\frac12)\) \(\approx\) \(1.282681243\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 - 4.33i)T \)
good7 \( 1 + (0.474 + 0.822i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (31.3 + 54.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (3.47 - 6.01i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 103.T + 4.91e3T^{2} \)
19 \( 1 - 73.8T + 6.85e3T^{2} \)
23 \( 1 + (43.0 - 74.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-27.7 - 48.1i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (99.9 - 173. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 18.9T + 5.06e4T^{2} \)
41 \( 1 + (-28.6 + 49.6i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (148. + 256. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (28.7 + 49.8i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 672.T + 1.48e5T^{2} \)
59 \( 1 + (-273. + 473. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-395. - 685. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (315. - 546. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 102.T + 3.57e5T^{2} \)
73 \( 1 + 200.T + 3.89e5T^{2} \)
79 \( 1 + (-332. - 576. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (673. + 1.16e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 12.6T + 7.04e5T^{2} \)
97 \( 1 + (-318. - 551. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−8.827715308531788260263026170632, −8.532673551169820812859969693728, −7.42074527585852455484722255401, −6.84123368010538358418103394175, −5.77320018514815309884979741130, −5.16164408896693348227075018395, −3.90722852022687524486439232923, −3.15265048477469347644043302084, −2.15786633557770296526903859932, −0.67108859835205024997994071377, 0.41393771995201027900443147540, 1.90495723718039919620474887513, 2.69434077093404090229254989490, 4.11521087281653779106786251251, 4.70281009743472236594115807536, 5.56026243713282026416780953383, 6.63664412319909321627889003194, 7.42456108328511366363906004497, 8.059898768061600190828215167061, 8.995364873162528452821417942766

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