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Properties

Label 2-1620-45.14-c2-0-2
Degree $2$
Conductor $1620$
Sign $-0.706 - 0.707i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.98 + 0.431i)5-s + (6.19 − 3.57i)7-s + (4.39 − 2.53i)11-s + (−2.70 − 1.56i)13-s − 8.72·17-s − 20.1·19-s + (−7.36 + 12.7i)23-s + (24.6 − 4.29i)25-s + (−34.4 + 19.8i)29-s + (19.6 − 34.0i)31-s + (−29.3 + 20.4i)35-s + 34.8i·37-s + (11.4 + 6.63i)41-s + (57.6 − 33.3i)43-s + (−8.45 − 14.6i)47-s + ⋯
L(s)  = 1  + (−0.996 + 0.0862i)5-s + (0.885 − 0.511i)7-s + (0.399 − 0.230i)11-s + (−0.208 − 0.120i)13-s − 0.513·17-s − 1.06·19-s + (−0.320 + 0.554i)23-s + (0.985 − 0.171i)25-s + (−1.18 + 0.686i)29-s + (0.635 − 1.09i)31-s + (−0.837 + 0.585i)35-s + 0.942i·37-s + (0.280 + 0.161i)41-s + (1.34 − 0.774i)43-s + (−0.179 − 0.311i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.706 - 0.707i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.706 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4393898540\)
\(L(\frac12)\) \(\approx\) \(0.4393898540\)
\(L(2)\) 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 \( 1 + (4.98 - 0.431i)T \)
good7 \( 1 + (-6.19 + 3.57i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-4.39 + 2.53i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (2.70 + 1.56i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 8.72T + 289T^{2} \)
19 \( 1 + 20.1T + 361T^{2} \)
23 \( 1 + (7.36 - 12.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (34.4 - 19.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-19.6 + 34.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 34.8iT - 1.36e3T^{2} \)
41 \( 1 + (-11.4 - 6.63i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-57.6 + 33.3i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (8.45 + 14.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 4.62T + 2.80e3T^{2} \)
59 \( 1 + (-22.3 - 12.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.09 - 10.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (92.1 + 53.2i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 101. iT - 5.04e3T^{2} \)
73 \( 1 - 23.2iT - 5.32e3T^{2} \)
79 \( 1 + (-33.2 - 57.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (72.1 + 124. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 154. iT - 7.92e3T^{2} \)
97 \( 1 + (152. - 87.8i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.394736032062437392943975788354, −8.557252728171851951606304863178, −7.904505986563865316291071146408, −7.26877151718819195630296089152, −6.40586771034426281711377157212, −5.29645216393751920860874312103, −4.28279216671074612809256333968, −3.86629491812680110806226787298, −2.51309077277415345768404518314, −1.21178155839639261663075071651, 0.12547124811894676714906590408, 1.66809672148179132779352673041, 2.72353614691780506157304655737, 4.11333101097834960142555581717, 4.49385403156647205653784079603, 5.57569332991833697261451387045, 6.58066356461474500760360653518, 7.41926218052986108490960954894, 8.191273858496361056572414466755, 8.749437981988363911966981069319

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