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Properties

Label 2-1620-12.11-c1-0-25
Degree $2$
Conductor $1620$
Sign $-0.997 + 0.0692i$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.03 + 0.964i)2-s + (0.138 + 1.99i)4-s + i·5-s + 3.54i·7-s + (−1.78 + 2.19i)8-s + (−0.964 + 1.03i)10-s − 4.36·11-s + 4.76·13-s + (−3.42 + 3.66i)14-s + (−3.96 + 0.552i)16-s + 3.02i·17-s + 1.16i·19-s + (−1.99 + 0.138i)20-s + (−4.51 − 4.21i)22-s + 3.11·23-s + ⋯
L(s)  = 1  + (0.731 + 0.682i)2-s + (0.0692 + 0.997i)4-s + 0.447i·5-s + 1.34i·7-s + (−0.629 + 0.776i)8-s + (−0.305 + 0.326i)10-s − 1.31·11-s + 1.32·13-s + (−0.915 + 0.980i)14-s + (−0.990 + 0.138i)16-s + 0.732i·17-s + 0.266i·19-s + (−0.446 + 0.0309i)20-s + (−0.962 − 0.897i)22-s + 0.648·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.997 + 0.0692i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ -0.997 + 0.0692i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.969661598\)
\(L(\frac12)\) \(\approx\) \(1.969661598\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 - 0.964i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 - 3.54iT - 7T^{2} \)
11 \( 1 + 4.36T + 11T^{2} \)
13 \( 1 - 4.76T + 13T^{2} \)
17 \( 1 - 3.02iT - 17T^{2} \)
19 \( 1 - 1.16iT - 19T^{2} \)
23 \( 1 - 3.11T + 23T^{2} \)
29 \( 1 + 8.14iT - 29T^{2} \)
31 \( 1 + 4.79iT - 31T^{2} \)
37 \( 1 + 7.83T + 37T^{2} \)
41 \( 1 + 4.61iT - 41T^{2} \)
43 \( 1 - 8.70iT - 43T^{2} \)
47 \( 1 + 8.13T + 47T^{2} \)
53 \( 1 - 11.9iT - 53T^{2} \)
59 \( 1 - 0.684T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 - 7.95iT - 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 9.41T + 73T^{2} \)
79 \( 1 - 8.75iT - 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + 4.93iT - 89T^{2} \)
97 \( 1 + 6.64T + 97T^{2} \)
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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

−9.677876455315487857402296480351, −8.604866152617798420754956887661, −8.242942436194401743760412922531, −7.38457759513873709482168109036, −6.22647109440790276516750513083, −5.87836288668472917435888961268, −5.08397359401234278439439191391, −3.94261860488014295140743007812, −2.98845601000050087076299265952, −2.17153761868542668190897646246, 0.57758421035527287809658599789, 1.66888108450421833263211336018, 3.14265467029319864242349880923, 3.73701674290961530434690049725, 4.96330049431419473957220535531, 5.21729442337423099493757813632, 6.58168035138344284954709411941, 7.17523572855155239703420545397, 8.294799699780709106929396387762, 9.083595364975923842008330159783

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