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Properties

Label 2-1620-45.4-c1-0-19
Degree $2$
Conductor $1620$
Sign $0.883 + 0.468i$
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  + (−0.308 + 2.21i)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 + (−4.80 − 1.36i)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  + (−0.138 + 0.990i)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 + (−0.961 − 0.273i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.883 + 0.468i$
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: \((2,\ 1620,\ (\ :1/2),\ 0.883 + 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.006460925\)
\(L(\frac12)\) \(\approx\) \(2.006460925\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.308 - 2.21i)T \)
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

−9.438940493861205999329177611235, −8.214617736002358788336809983306, −7.79221965168752853927603853373, −7.08582315302168096674492957271, −6.16593414374674588494542409997, −5.11900568175005417097879115505, −4.30977830261075236567259466761, −3.34241832448399690583193090724, −2.24625598883564968524010604550, −0.881036648444606074505320985311, 1.36006580129881154618013618161, 2.04604359561509017836968457008, 3.65153620883119792429361298357, 4.59085622085063585179479381611, 5.42011565525888850798458715243, 5.79747520875506572663816968068, 7.33477904992382602103965853995, 8.083304949344081096408174177672, 8.563756566074263210810915578120, 9.177721445304902653742783548725

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