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Properties

Label 2-1620-12.11-c1-0-26
Degree $2$
Conductor $1620$
Sign $-0.254 - 0.967i$
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  + (−0.863 + 1.12i)2-s + (−0.509 − 1.93i)4-s + i·5-s − 1.02i·7-s + (2.60 + 1.09i)8-s + (−1.12 − 0.863i)10-s + 3.69·11-s − 5.05·13-s + (1.15 + 0.889i)14-s + (−3.48 + 1.97i)16-s − 2.05i·17-s + 5.65i·19-s + (1.93 − 0.509i)20-s + (−3.19 + 4.14i)22-s − 0.311·23-s + ⋯
L(s)  = 1  + (−0.610 + 0.792i)2-s + (−0.254 − 0.967i)4-s + 0.447i·5-s − 0.389i·7-s + (0.921 + 0.388i)8-s + (−0.354 − 0.272i)10-s + 1.11·11-s − 1.40·13-s + (0.308 + 0.237i)14-s + (−0.870 + 0.492i)16-s − 0.497i·17-s + 1.29i·19-s + (0.432 − 0.113i)20-s + (−0.680 + 0.883i)22-s − 0.0648·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.254 - 0.967i$
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: \((2,\ 1620,\ (\ :1/2),\ -0.254 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.043198599\)
\(L(\frac12)\) \(\approx\) \(1.043198599\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.863 - 1.12i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + 1.02iT - 7T^{2} \)
11 \( 1 - 3.69T + 11T^{2} \)
13 \( 1 + 5.05T + 13T^{2} \)
17 \( 1 + 2.05iT - 17T^{2} \)
19 \( 1 - 5.65iT - 19T^{2} \)
23 \( 1 + 0.311T + 23T^{2} \)
29 \( 1 + 0.993iT - 29T^{2} \)
31 \( 1 - 4.07iT - 31T^{2} \)
37 \( 1 - 8.38T + 37T^{2} \)
41 \( 1 - 9.77iT - 41T^{2} \)
43 \( 1 + 4.84iT - 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 5.37iT - 53T^{2} \)
59 \( 1 - 8.13T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 + 4.45iT - 67T^{2} \)
71 \( 1 + 0.205T + 71T^{2} \)
73 \( 1 + 5.22T + 73T^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 - 0.215T + 83T^{2} \)
89 \( 1 - 18.2iT - 89T^{2} \)
97 \( 1 - 9.67T + 97T^{2} \)
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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

−9.606681905790797471258943903424, −8.869346048243782881714094655694, −7.80039995834692550396522766322, −7.33842559863560164457958008494, −6.54337987275036420267536403243, −5.81089787004989659670208135949, −4.75875266948244679181416552354, −3.90418225475206203133438990772, −2.44804031173778593884265633574, −1.11275110502618587844751027128, 0.58394929382502434813287236119, 1.95301709869737359329816221267, 2.82427823336621879465514048792, 4.08280172098881164243859122259, 4.71252529594205768665233799416, 5.89657487011695640998634269536, 7.05694438625943167943857994721, 7.64335185635125781781609322072, 8.772148702532980127992478178552, 9.125295140179498081419768244892

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