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Properties

Label 2-1620-45.4-c1-0-12
Degree $2$
Conductor $1620$
Sign $0.726 - 0.687i$
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  + (1.23 + 1.86i)5-s + (3.46 − 2i)7-s + (2 + 3.46i)11-s + 4i·17-s + (3.46 + 2i)23-s + (−1.96 + 4.59i)25-s + (−3 − 5.19i)29-s + (−2 + 3.46i)31-s + (8 + 4i)35-s − 8i·37-s + (5 − 8.66i)41-s + (−3.46 + 2i)43-s + (−3.46 + 2i)47-s + (4.49 − 7.79i)49-s + 12i·53-s + ⋯
L(s)  = 1  + (0.550 + 0.834i)5-s + (1.30 − 0.755i)7-s + (0.603 + 1.04i)11-s + 0.970i·17-s + (0.722 + 0.417i)23-s + (−0.392 + 0.919i)25-s + (−0.557 − 0.964i)29-s + (−0.359 + 0.622i)31-s + (1.35 + 0.676i)35-s − 1.31i·37-s + (0.780 − 1.35i)41-s + (−0.528 + 0.304i)43-s + (−0.505 + 0.291i)47-s + (0.642 − 1.11i)49-s + 1.64i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.726 - 0.687i$
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: \((2,\ 1620,\ (\ :1/2),\ 0.726 - 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.285098398\)
\(L(\frac12)\) \(\approx\) \(2.285098398\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 1.86i)T \)
good7 \( 1 + (-3.46 + 2i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-3.46 - 2i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + (-5 + 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.46 - 2i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.46 - 2i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.46 - 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.46 - 2i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-6.92 + 4i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.556100248431572131727442606771, −8.753917271321828150628947815120, −7.56916942861005585525355350340, −7.32278411839072695737823594711, −6.31445654900072427012460982550, −5.40338262845794400862969989394, −4.41055088770428777547438274640, −3.66847953874837885860324728168, −2.20587067679222812709681630030, −1.44872113748713862608872106888, 1.00022395715148385726328433653, 1.99758018313292688070117116719, 3.18987741052472924101280296814, 4.57599004863505965373510384617, 5.15051545039247505311528480346, 5.85522153144080576611243305418, 6.82093298763815360226846357120, 8.009004515271222247859368282561, 8.567472859117818704247847468861, 9.124410620979882437952675262575

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