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Properties

Label 2-1620-45.34-c1-0-21
Degree $2$
Conductor $1620$
Sign $-0.996 + 0.0871i$
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  + (−2.15 − 0.578i)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  + (−0.965 − 0.258i)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

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

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.996 + 0.0871i$
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: \((2,\ 1620,\ (\ :1/2),\ -0.996 + 0.0871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09805570402\)
\(L(\frac12)\) \(\approx\) \(0.09805570402\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 + 0.578i)T \)
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

−8.750135015569744607075949452593, −8.350839637452861861132377420995, −7.32373120404719408385345912892, −6.91796410704114688502231566966, −5.46479076346000552512276870357, −4.91983428928989338992640823576, −3.92504335828855760590053390098, −2.99956394441956712294455208711, −1.66181054432530440957269599215, −0.03742473392873267428236626882, 1.62078525705690795080572163916, 3.02098029914625644917757263399, 3.96514174262278971480824297609, 4.53880469050416269607574615166, 5.98534483235687590605683859473, 6.38455479319136673529196433460, 7.65344949250391293689302838217, 8.178235794151910097575843912518, 8.671766962980003505235794229540, 9.884088221587052353127510581576

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