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Properties

Label 2-1620-12.11-c1-0-10
Degree $2$
Conductor $1620$
Sign $0.847 - 0.530i$
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  + (−0.390 − 1.35i)2-s + (−1.69 + 1.06i)4-s − i·5-s − 1.67i·7-s + (2.10 + 1.89i)8-s + (−1.35 + 0.390i)10-s − 4.67·11-s − 0.142·13-s + (−2.27 + 0.653i)14-s + (1.74 − 3.59i)16-s + 7.09i·17-s − 0.158i·19-s + (1.06 + 1.69i)20-s + (1.82 + 6.35i)22-s + 0.185·23-s + ⋯
L(s)  = 1  + (−0.276 − 0.961i)2-s + (−0.847 + 0.530i)4-s − 0.447i·5-s − 0.632i·7-s + (0.743 + 0.668i)8-s + (−0.429 + 0.123i)10-s − 1.41·11-s − 0.0394·13-s + (−0.607 + 0.174i)14-s + (0.436 − 0.899i)16-s + 1.72i·17-s − 0.0363i·19-s + (0.237 + 0.379i)20-s + (0.389 + 1.35i)22-s + 0.0386·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.847 - 0.530i$
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: \((2,\ 1620,\ (\ :1/2),\ 0.847 - 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6343820224\)
\(L(\frac12)\) \(\approx\) \(0.6343820224\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.390 + 1.35i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 + 1.67iT - 7T^{2} \)
11 \( 1 + 4.67T + 11T^{2} \)
13 \( 1 + 0.142T + 13T^{2} \)
17 \( 1 - 7.09iT - 17T^{2} \)
19 \( 1 + 0.158iT - 19T^{2} \)
23 \( 1 - 0.185T + 23T^{2} \)
29 \( 1 - 3.22iT - 29T^{2} \)
31 \( 1 + 5.91iT - 31T^{2} \)
37 \( 1 - 0.634T + 37T^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 - 8.39iT - 43T^{2} \)
47 \( 1 + 9.55T + 47T^{2} \)
53 \( 1 + 7.27iT - 53T^{2} \)
59 \( 1 - 6.46T + 59T^{2} \)
61 \( 1 - 6.27T + 61T^{2} \)
67 \( 1 - 8.44iT - 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 + 4.30T + 73T^{2} \)
79 \( 1 - 13.8iT - 79T^{2} \)
83 \( 1 - 6.56T + 83T^{2} \)
89 \( 1 - 11.0iT - 89T^{2} \)
97 \( 1 - 3.53T + 97T^{2} \)
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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

−9.830438544812762953619636127469, −8.569056009931895086427502443007, −8.147590567198068592206063579965, −7.41189927886537412782295293212, −6.10169537107320219813841122870, −5.09147780559062669297112691557, −4.31923981388377092802362465331, −3.42105807882707493747440273803, −2.32505714422599336994335223876, −1.17631377984330585522553049834, 0.29522174711573040022899717031, 2.23814341200885015185223591826, 3.28448697429362753951398612767, 4.71356198423455333608449812771, 5.33115915258786678316724049695, 6.05617047745555559016242934106, 7.19312679449028668057349471737, 7.47971115080671305933389808027, 8.545002725195166026030276459516, 9.095967093689370777206602025078

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