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Properties

Label 2-1620-12.11-c1-0-16
Degree $2$
Conductor $1620$
Sign $-0.408 - 0.912i$
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  + (−1.38 + 0.295i)2-s + (1.82 − 0.817i)4-s + i·5-s − 1.21i·7-s + (−2.28 + 1.66i)8-s + (−0.295 − 1.38i)10-s − 3.73·11-s + 1.24·13-s + (0.360 + 1.68i)14-s + (2.66 − 2.98i)16-s − 0.279i·17-s − 0.369i·19-s + (0.817 + 1.82i)20-s + (5.16 − 1.10i)22-s + 3.54·23-s + ⋯
L(s)  = 1  + (−0.977 + 0.208i)2-s + (0.912 − 0.408i)4-s + 0.447i·5-s − 0.460i·7-s + (−0.807 + 0.590i)8-s + (−0.0934 − 0.437i)10-s − 1.12·11-s + 0.345·13-s + (0.0962 + 0.450i)14-s + (0.666 − 0.745i)16-s − 0.0678i·17-s − 0.0848i·19-s + (0.182 + 0.408i)20-s + (1.10 − 0.235i)22-s + 0.739·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.408 - 0.912i$
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: \((2,\ 1620,\ (\ :1/2),\ -0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6342191625\)
\(L(\frac12)\) \(\approx\) \(0.6342191625\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.295i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + 1.21iT - 7T^{2} \)
11 \( 1 + 3.73T + 11T^{2} \)
13 \( 1 - 1.24T + 13T^{2} \)
17 \( 1 + 0.279iT - 17T^{2} \)
19 \( 1 + 0.369iT - 19T^{2} \)
23 \( 1 - 3.54T + 23T^{2} \)
29 \( 1 - 1.80iT - 29T^{2} \)
31 \( 1 - 6.56iT - 31T^{2} \)
37 \( 1 + 7.25T + 37T^{2} \)
41 \( 1 - 11.1iT - 41T^{2} \)
43 \( 1 - 0.149iT - 43T^{2} \)
47 \( 1 + 7.93T + 47T^{2} \)
53 \( 1 - 3.61iT - 53T^{2} \)
59 \( 1 - 8.63T + 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + 1.29iT - 67T^{2} \)
71 \( 1 + 0.390T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 14.8iT - 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 - 12.9iT - 89T^{2} \)
97 \( 1 + 1.66T + 97T^{2} \)
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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

−9.778175794301639144243851015798, −8.684240567872588698663732305923, −8.178954119136962564797969020310, −7.19440890522707584503797257243, −6.80923599446611149138602898744, −5.71850413091138343920910094457, −4.89766206056030471553899718699, −3.39834782445694473284883808093, −2.55667010612075169295928846304, −1.22468332629215593003715314186, 0.35542550277957460333495487629, 1.84017151643605102523118578063, 2.76599011569974011661377883264, 3.86926930047159581456318495373, 5.21432633423217442796596267055, 5.90107132074643214141642432526, 6.99654748739433356861623759085, 7.71725041088482370063244748146, 8.566965831182044561778608385751, 8.945117872268385714894017554684

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