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Properties

Label 2-1620-45.34-c1-0-3
Degree $2$
Conductor $1620$
Sign $-0.0871 - 0.996i$
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  + (0.578 − 2.15i)5-s + (−0.866 − 0.5i)7-s + (−1.58 + 2.73i)11-s + (−2.59 + 1.5i)13-s + 6.32i·17-s − 3·19-s + (2.73 − 1.58i)23-s + (−4.33 − 2.5i)25-s + (−4.74 + 8.21i)29-s + (1 + 1.73i)31-s + (−1.58 + 1.58i)35-s + i·37-s + (−1.58 − 2.73i)41-s + (8.66 + 5i)43-s + (5.47 + 3.16i)47-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)5-s + (−0.327 − 0.188i)7-s + (−0.476 + 0.825i)11-s + (−0.720 + 0.416i)13-s + 1.53i·17-s − 0.688·19-s + (0.571 − 0.329i)23-s + (−0.866 − 0.5i)25-s + (−0.880 + 1.52i)29-s + (0.179 + 0.311i)31-s + (−0.267 + 0.267i)35-s + 0.164i·37-s + (−0.246 − 0.427i)41-s + (1.32 + 0.762i)43-s + (0.798 + 0.461i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.0871 - 0.996i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ -0.0871 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8875808594\)
\(L(\frac12)\) \(\approx\) \(0.8875808594\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.578 + 2.15i)T \)
good7 \( 1 + (0.866 + 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.58 - 2.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.59 - 1.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.32iT - 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + (-2.73 + 1.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.74 - 8.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - iT - 37T^{2} \)
41 \( 1 + (1.58 + 2.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.66 - 5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.47 - 3.16i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.48iT - 53T^{2} \)
59 \( 1 + (-3.16 - 5.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.52 + 5.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.48T + 71T^{2} \)
73 \( 1 + 13iT - 73T^{2} \)
79 \( 1 + (1.5 - 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-13.6 - 7.90i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (-0.866 - 0.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.462771608621529208746437618567, −8.928666117489382026243064079942, −8.068476443963350302944770287262, −7.23905885462716958578101372608, −6.37063483794953723466019350594, −5.42202199747375061378016562638, −4.63652937578857941265121117651, −3.87179569316202417784372084863, −2.43475608384686123739468764718, −1.42912919320101596940024571778, 0.33166362407911069372282534908, 2.37172855718891702475218011160, 2.89876140040363285871116094943, 3.99202946385166518186139732332, 5.28238901844491945145063862082, 5.87691697478407096698154497648, 6.85882407880253252892622947088, 7.47690655282554498311263945459, 8.317285966129974945944748059616, 9.439837879407127594868679203502

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