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Properties

Label 2-1620-5.3-c2-0-47
Degree $2$
Conductor $1620$
Sign $-0.786 - 0.616i$
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  + (0.555 − 4.96i)5-s + (3.23 − 3.23i)7-s − 6.14·11-s + (−16.3 − 16.3i)13-s + (6.88 − 6.88i)17-s + 20.9i·19-s + (−25.3 − 25.3i)23-s + (−24.3 − 5.51i)25-s + 39.5i·29-s + 15.0·31-s + (−14.2 − 17.8i)35-s + (−47.5 + 47.5i)37-s + 31.1·41-s + (6.13 + 6.13i)43-s + (13.3 − 13.3i)47-s + ⋯
L(s)  = 1  + (0.111 − 0.993i)5-s + (0.462 − 0.462i)7-s − 0.558·11-s + (−1.25 − 1.25i)13-s + (0.405 − 0.405i)17-s + 1.10i·19-s + (−1.10 − 1.10i)23-s + (−0.975 − 0.220i)25-s + 1.36i·29-s + 0.485·31-s + (−0.407 − 0.510i)35-s + (−1.28 + 1.28i)37-s + 0.759·41-s + (0.142 + 0.142i)43-s + (0.283 − 0.283i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 - 0.616i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.786 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.786 - 0.616i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.786 - 0.616i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2900632551\)
\(L(\frac12)\) \(\approx\) \(0.2900632551\)
\(L(2)\) 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 \( 1 + (-0.555 + 4.96i)T \)
good7 \( 1 + (-3.23 + 3.23i)T - 49iT^{2} \)
11 \( 1 + 6.14T + 121T^{2} \)
13 \( 1 + (16.3 + 16.3i)T + 169iT^{2} \)
17 \( 1 + (-6.88 + 6.88i)T - 289iT^{2} \)
19 \( 1 - 20.9iT - 361T^{2} \)
23 \( 1 + (25.3 + 25.3i)T + 529iT^{2} \)
29 \( 1 - 39.5iT - 841T^{2} \)
31 \( 1 - 15.0T + 961T^{2} \)
37 \( 1 + (47.5 - 47.5i)T - 1.36e3iT^{2} \)
41 \( 1 - 31.1T + 1.68e3T^{2} \)
43 \( 1 + (-6.13 - 6.13i)T + 1.84e3iT^{2} \)
47 \( 1 + (-13.3 + 13.3i)T - 2.20e3iT^{2} \)
53 \( 1 + (58.4 + 58.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 8.96iT - 3.48e3T^{2} \)
61 \( 1 + 10.8T + 3.72e3T^{2} \)
67 \( 1 + (24.3 - 24.3i)T - 4.48e3iT^{2} \)
71 \( 1 - 89.8T + 5.04e3T^{2} \)
73 \( 1 + (-42.9 - 42.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 35.9iT - 6.24e3T^{2} \)
83 \( 1 + (-31.5 - 31.5i)T + 6.88e3iT^{2} \)
89 \( 1 - 75.7iT - 7.92e3T^{2} \)
97 \( 1 + (21.5 - 21.5i)T - 9.40e3iT^{2} \)
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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

−8.481120431965093064261535122495, −8.031585076177793632738748289316, −7.37943272348993920886030663776, −6.16436777173539431706219846894, −5.14911020081748119297033210616, −4.84557431724252587449967024371, −3.63734127392011131162189992049, −2.46059526996956512157716477106, −1.23819092823455484381387624938, −0.07638982570281666118861500844, 1.98152060268581153858616090908, 2.52766135586864377157347273682, 3.77470546714303883905433906327, 4.75104222623885376377317016921, 5.67540332086955093062044275390, 6.48973343036176770322574736845, 7.42973914838199131701277985316, 7.83307949961617263385459474437, 9.066812151697201729096508949542, 9.665904222181137590661309775459

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