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Properties

Label 2-1620-9.7-c3-0-43
Degree $2$
Conductor $1620$
Sign $-0.642 + 0.766i$
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 + (5.24 − 9.07i)7-s + (16.6 − 28.8i)11-s + (11.9 + 20.7i)13-s + 72.5·17-s − 45.9·19-s + (−81.5 − 141. i)23-s + (−12.5 + 21.6i)25-s + (−15.4 + 26.8i)29-s + (−152. − 263. i)31-s − 52.4·35-s + 150.·37-s + (204. + 354. i)41-s + (124. − 215. i)43-s + (77.9 − 135. i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.282 − 0.490i)7-s + (0.456 − 0.791i)11-s + (0.255 + 0.442i)13-s + 1.03·17-s − 0.555·19-s + (−0.739 − 1.28i)23-s + (−0.100 + 0.173i)25-s + (−0.0991 + 0.171i)29-s + (−0.882 − 1.52i)31-s − 0.253·35-s + 0.667·37-s + (0.779 + 1.34i)41-s + (0.441 − 0.765i)43-s + (0.241 − 0.419i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.607818322\)
\(L(\frac12)\) \(\approx\) \(1.607818322\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.24 + 9.07i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-16.6 + 28.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-11.9 - 20.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 72.5T + 4.91e3T^{2} \)
19 \( 1 + 45.9T + 6.85e3T^{2} \)
23 \( 1 + (81.5 + 141. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (15.4 - 26.8i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (152. + 263. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 150.T + 5.06e4T^{2} \)
41 \( 1 + (-204. - 354. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-124. + 215. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-77.9 + 135. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 263.T + 1.48e5T^{2} \)
59 \( 1 + (-122. - 212. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-233. + 403. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-88.5 - 153. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 45.2T + 3.57e5T^{2} \)
73 \( 1 + 949.T + 3.89e5T^{2} \)
79 \( 1 + (248. - 430. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (177. - 307. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.19e3T + 7.04e5T^{2} \)
97 \( 1 + (-204. + 354. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−8.648203221739781962605401204216, −8.036922298519879356694526219313, −7.23838781672640310164571527608, −6.21246199620451052460630690013, −5.58370811668812284453912325661, −4.29413465633777271717149298113, −3.91801705603761851712221063819, −2.59601750005731361265944340294, −1.32350120491914225093296143229, −0.37523058190476067226631585658, 1.24719686508712268388176475774, 2.28001886148146404369786078353, 3.41243320115388669526305473530, 4.20314139752482058207984243531, 5.36797075490827791990115202471, 5.95636940374330251113057149964, 7.08786263095634156686171048672, 7.63335638261460665153123671532, 8.516658477849726813577551782060, 9.320099387657205114901451676359

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