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Properties

Label 2-1620-12.11-c1-0-24
Degree $2$
Conductor $1620$
Sign $0.936 - 0.349i$
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  + (−1.16 − 0.806i)2-s + (0.699 + 1.87i)4-s − i·5-s − 1.36i·7-s + (0.697 − 2.74i)8-s + (−0.806 + 1.16i)10-s + 4.01·11-s − 5.78·13-s + (−1.10 + 1.58i)14-s + (−3.02 + 2.62i)16-s + 6.59i·17-s + 6.69i·19-s + (1.87 − 0.699i)20-s + (−4.66 − 3.23i)22-s + 6.03·23-s + ⋯
L(s)  = 1  + (−0.821 − 0.570i)2-s + (0.349 + 0.936i)4-s − 0.447i·5-s − 0.516i·7-s + (0.246 − 0.969i)8-s + (−0.254 + 0.367i)10-s + 1.21·11-s − 1.60·13-s + (−0.294 + 0.424i)14-s + (−0.755 + 0.655i)16-s + 1.59i·17-s + 1.53i·19-s + (0.418 − 0.156i)20-s + (−0.995 − 0.690i)22-s + 1.25·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.936 - 0.349i$
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: \((2,\ 1620,\ (\ :1/2),\ 0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9604341318\)
\(L(\frac12)\) \(\approx\) \(0.9604341318\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 0.806i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 + 1.36iT - 7T^{2} \)
11 \( 1 - 4.01T + 11T^{2} \)
13 \( 1 + 5.78T + 13T^{2} \)
17 \( 1 - 6.59iT - 17T^{2} \)
19 \( 1 - 6.69iT - 19T^{2} \)
23 \( 1 - 6.03T + 23T^{2} \)
29 \( 1 + 3.39iT - 29T^{2} \)
31 \( 1 - 8.20iT - 31T^{2} \)
37 \( 1 + 5.39T + 37T^{2} \)
41 \( 1 + 1.20iT - 41T^{2} \)
43 \( 1 + 4.11iT - 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 - 0.306iT - 53T^{2} \)
59 \( 1 - 5.52T + 59T^{2} \)
61 \( 1 - 5.15T + 61T^{2} \)
67 \( 1 - 10.2iT - 67T^{2} \)
71 \( 1 - 2.50T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 - 6.00iT - 79T^{2} \)
83 \( 1 - 9.38T + 83T^{2} \)
89 \( 1 - 15.2iT - 89T^{2} \)
97 \( 1 - 3.45T + 97T^{2} \)
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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

−9.541967675794541768964995146734, −8.667082462343979675539304890690, −8.095121216643205517905161193714, −7.13588560592829924248898368999, −6.54763794302852690830949371620, −5.24000276936538972187812166074, −4.10276228865541363515114595674, −3.49154757995578497718455387240, −2.03813967520565785148877080278, −1.13641294848701714489553371125, 0.55602806917560942124540122898, 2.17773018901198038594666209948, 3.00155974801695754212325627238, 4.75003354652324076993164202793, 5.21967063144871310211989903712, 6.50490374623839898250402885135, 6.99984521441150263006845030674, 7.56288686618681341295579978442, 8.735413895817253796396203388730, 9.483018146813861435129147756721

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