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Properties

Label 2-1620-12.11-c1-0-28
Degree $2$
Conductor $1620$
Sign $-0.994 - 0.103i$
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  + (−0.0733 + 1.41i)2-s + (−1.98 − 0.207i)4-s + i·5-s + 4.10i·7-s + (0.438 − 2.79i)8-s + (−1.41 − 0.0733i)10-s + 2.57·11-s + 2.47·13-s + (−5.79 − 0.301i)14-s + (3.91 + 0.824i)16-s + 5.59i·17-s + 0.255i·19-s + (0.207 − 1.98i)20-s + (−0.188 + 3.63i)22-s + 7.17·23-s + ⋯
L(s)  = 1  + (−0.0518 + 0.998i)2-s + (−0.994 − 0.103i)4-s + 0.447i·5-s + 1.55i·7-s + (0.155 − 0.987i)8-s + (−0.446 − 0.0231i)10-s + 0.776·11-s + 0.687·13-s + (−1.54 − 0.0804i)14-s + (0.978 + 0.206i)16-s + 1.35i·17-s + 0.0586i·19-s + (0.0463 − 0.444i)20-s + (−0.0402 + 0.775i)22-s + 1.49·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.994 - 0.103i$
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: \((2,\ 1620,\ (\ :1/2),\ -0.994 - 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.463868678\)
\(L(\frac12)\) \(\approx\) \(1.463868678\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0733 - 1.41i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 - 4.10iT - 7T^{2} \)
11 \( 1 - 2.57T + 11T^{2} \)
13 \( 1 - 2.47T + 13T^{2} \)
17 \( 1 - 5.59iT - 17T^{2} \)
19 \( 1 - 0.255iT - 19T^{2} \)
23 \( 1 - 7.17T + 23T^{2} \)
29 \( 1 - 5.52iT - 29T^{2} \)
31 \( 1 + 9.47iT - 31T^{2} \)
37 \( 1 + 5.63T + 37T^{2} \)
41 \( 1 - 4.21iT - 41T^{2} \)
43 \( 1 - 2.94iT - 43T^{2} \)
47 \( 1 - 3.93T + 47T^{2} \)
53 \( 1 - 4.80iT - 53T^{2} \)
59 \( 1 - 0.827T + 59T^{2} \)
61 \( 1 + 4.94T + 61T^{2} \)
67 \( 1 + 8.67iT - 67T^{2} \)
71 \( 1 + 8.80T + 71T^{2} \)
73 \( 1 + 1.18T + 73T^{2} \)
79 \( 1 + 4.47iT - 79T^{2} \)
83 \( 1 + 7.30T + 83T^{2} \)
89 \( 1 + 6.33iT - 89T^{2} \)
97 \( 1 + 0.862T + 97T^{2} \)
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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

−9.335462467323425349171571239803, −8.920390050740126737470924428207, −8.283145938666103938890014767498, −7.33170271676627891287961296758, −6.32121450838367322906900165862, −6.01064023566484465553980579275, −5.09644405279804938682110682269, −3.99735306300870555398853212570, −3.03153403123305611988287428454, −1.54208493094909718849516894339, 0.66252369324973846669123078182, 1.44566084731649487528894623189, 3.01544812322713858378884735441, 3.85466103438814468204630165889, 4.57838037336421321144848752023, 5.39955134929264484944899086414, 6.81914413621904569364913370915, 7.39267774590441594570530372192, 8.557245596314140816877280459387, 9.068880404123642548787738592357

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