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Properties

Label 2-1620-45.4-c1-0-10
Degree $2$
Conductor $1620$
Sign $0.308 + 0.951i$
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  + (−1.76 + 1.37i)5-s + (−4.27 + 2.46i)7-s + (−1.20 − 2.08i)11-s + (−2.51 − 1.45i)13-s + 6.86i·17-s + 4.17·19-s + (2.90 + 1.67i)23-s + (1.21 − 4.84i)25-s + (−2.59 − 4.5i)29-s + (3.08 − 5.35i)31-s + (4.14 − 10.2i)35-s − 7.84i·37-s + (−2.93 + 5.08i)41-s + (4.27 − 2.46i)43-s + (−10.3 + 5.95i)47-s + ⋯
L(s)  = 1  + (−0.788 + 0.614i)5-s + (−1.61 + 0.932i)7-s + (−0.363 − 0.629i)11-s + (−0.698 − 0.403i)13-s + 1.66i·17-s + 0.958·19-s + (0.606 + 0.350i)23-s + (0.243 − 0.969i)25-s + (−0.482 − 0.835i)29-s + (0.554 − 0.961i)31-s + (0.700 − 1.72i)35-s − 1.28i·37-s + (−0.458 + 0.794i)41-s + (0.651 − 0.376i)43-s + (−1.50 + 0.868i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.308 + 0.951i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.308 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4888686157\)
\(L(\frac12)\) \(\approx\) \(0.4888686157\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.76 - 1.37i)T \)
good7 \( 1 + (4.27 - 2.46i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.20 + 2.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.51 + 1.45i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.86iT - 17T^{2} \)
19 \( 1 - 4.17T + 19T^{2} \)
23 \( 1 + (-2.90 - 1.67i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.59 + 4.5i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.08 + 5.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.84iT - 37T^{2} \)
41 \( 1 + (2.93 - 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.27 + 2.46i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (10.3 - 5.95i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.54iT - 53T^{2} \)
59 \( 1 + (-0.525 + 0.910i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.58 + 7.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.50 - 2.02i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 - 2.02iT - 73T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.49 + 2.59i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.09T + 89T^{2} \)
97 \( 1 + (0.764 - 0.441i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.432070074812100458099212732306, −8.291054269297782321918828246950, −7.76011545768212345807224310110, −6.69414217771756331840444546470, −6.09416745332009173903285766385, −5.30738734611490887675758170358, −3.84130815877934328531724906918, −3.23030815872220211082799350046, −2.41472202675914516070742932396, −0.23843110766605416678379020671, 0.939709535878905459624224337833, 2.84489706000344997380289350819, 3.51416631361347990535560160378, 4.65344448760350769121694363176, 5.17140820396978133566613257641, 6.70817499689037999276100128372, 7.10945870623945892798976311652, 7.73386960770610246922395582104, 8.980505670257426556302607165604, 9.541014764855542975324277913552

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