Properties

Label 2-1620-45.34-c1-0-23
Degree $2$
Conductor $1620$
Sign $-0.991 - 0.127i$
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  + (1.02 − 1.98i)5-s + (1.10 + 0.640i)7-s + (−2.07 + 3.58i)11-s + (−5.64 + 3.26i)13-s − 5.98i·17-s − 7.17·19-s + (−6.52 + 3.76i)23-s + (−2.88 − 4.08i)25-s + (2.59 − 4.5i)29-s + (−2.58 − 4.48i)31-s + (2.41 − 1.54i)35-s + 5.24i·37-s + (−0.340 − 0.589i)41-s + (−1.10 − 0.640i)43-s + (4.59 + 2.65i)47-s + ⋯
L(s)  = 1  + (0.459 − 0.888i)5-s + (0.419 + 0.242i)7-s + (−0.624 + 1.08i)11-s + (−1.56 + 0.904i)13-s − 1.45i·17-s − 1.64·19-s + (−1.35 + 0.785i)23-s + (−0.577 − 0.816i)25-s + (0.482 − 0.835i)29-s + (−0.465 − 0.805i)31-s + (0.407 − 0.261i)35-s + 0.861i·37-s + (−0.0531 − 0.0920i)41-s + (−0.169 − 0.0977i)43-s + (0.670 + 0.387i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.991 - 0.127i$
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: \((2,\ 1620,\ (\ :1/2),\ -0.991 - 0.127i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05822213080\)
\(L(\frac12)\) \(\approx\) \(0.05822213080\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.02 + 1.98i)T \)
good7 \( 1 + (-1.10 - 0.640i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.07 - 3.58i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.64 - 3.26i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.98iT - 17T^{2} \)
19 \( 1 + 7.17T + 19T^{2} \)
23 \( 1 + (6.52 - 3.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.59 + 4.5i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.58 + 4.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.24iT - 37T^{2} \)
41 \( 1 + (0.340 + 0.589i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.10 + 0.640i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.59 - 2.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.21iT - 53T^{2} \)
59 \( 1 + (3.80 + 6.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.08 + 1.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.5 - 7.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.50T + 71T^{2} \)
73 \( 1 - 7.80iT - 73T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.44 - 4.87i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 + (12.4 + 7.16i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.154773702330398117716450918616, −8.129200568160397990487354403063, −7.50413210237387265129665340626, −6.58707229635040832407215324287, −5.52509713374600994385301795668, −4.71220810852453134667810598204, −4.33098214148549247785365769799, −2.38523101530783646400627104837, −1.96872348978119078149566300262, −0.01959595748559724578333426395, 1.95117638423494783475785305635, 2.78782745505962000375495597297, 3.82962378610035612905763675710, 4.93435336974575591729235517083, 5.88708406211934328232050167250, 6.45371702974103520715221144548, 7.53789333261221031445287621664, 8.125679896959947727245642574179, 8.909730531898296975765406746049, 10.14131292274926753099917247364

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