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Properties

Label 2-3620-3620.2259-c0-0-0
Degree $2$
Conductor $3620$
Sign $0.994 - 0.103i$
Analytic cond. $1.80661$
Root an. cond. $1.34410$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.241 − 0.970i)2-s + (0.0916 + 0.187i)3-s + (−0.882 − 0.469i)4-s + (0.913 + 0.406i)5-s + (0.204 − 0.0434i)6-s + (−0.939 + 1.62i)7-s + (−0.669 + 0.743i)8-s + (0.588 − 0.753i)9-s + (0.615 − 0.788i)10-s + (0.00729 − 0.208i)12-s + (1.35 + 1.30i)14-s + (0.00729 + 0.208i)15-s + (0.559 + 0.829i)16-s + (−0.588 − 0.753i)18-s + (−0.615 − 0.788i)20-s + (−0.391 − 0.0274i)21-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + (0.0916 + 0.187i)3-s + (−0.882 − 0.469i)4-s + (0.913 + 0.406i)5-s + (0.204 − 0.0434i)6-s + (−0.939 + 1.62i)7-s + (−0.669 + 0.743i)8-s + (0.588 − 0.753i)9-s + (0.615 − 0.788i)10-s + (0.00729 − 0.208i)12-s + (1.35 + 1.30i)14-s + (0.00729 + 0.208i)15-s + (0.559 + 0.829i)16-s + (−0.588 − 0.753i)18-s + (−0.615 − 0.788i)20-s + (−0.391 − 0.0274i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3620\)    =    \(2^{2} \cdot 5 \cdot 181\)
Sign: $0.994 - 0.103i$
Analytic conductor: \(1.80661\)
Root analytic conductor: \(1.34410\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3620} (2259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3620,\ (\ :0),\ 0.994 - 0.103i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.404696162\)
\(L(\frac12)\) \(\approx\) \(1.404696162\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.241 + 0.970i)T \)
5 \( 1 + (-0.913 - 0.406i)T \)
181 \( 1 + (0.882 + 0.469i)T \)
good3 \( 1 + (-0.0916 - 0.187i)T + (-0.615 + 0.788i)T^{2} \)
7 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.438 + 0.898i)T^{2} \)
13 \( 1 + (-0.0348 + 0.999i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.328 + 0.812i)T + (-0.719 - 0.694i)T^{2} \)
29 \( 1 + (1.33 - 1.48i)T + (-0.104 - 0.994i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.990 - 0.139i)T^{2} \)
41 \( 1 + (-1.51 + 0.213i)T + (0.961 - 0.275i)T^{2} \)
43 \( 1 + (0.333 - 1.89i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.0467 - 1.33i)T + (-0.997 - 0.0697i)T^{2} \)
53 \( 1 + (-0.961 - 0.275i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-1.51 - 1.27i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-1.78 + 0.379i)T + (0.913 - 0.406i)T^{2} \)
71 \( 1 + (0.978 + 0.207i)T^{2} \)
73 \( 1 + (-0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.961 - 0.275i)T^{2} \)
83 \( 1 + (0.594 - 0.170i)T + (0.848 - 0.529i)T^{2} \)
89 \( 1 + (1.35 - 1.13i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.438 - 0.898i)T^{2} \)
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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

−9.092162449748914845349878179466, −8.458224195886942746284633351944, −7.01247013825156377051622610964, −6.24583018931208733467093630199, −5.72604113843547296036412573858, −4.97753166577419377506829705558, −3.86432813729848408617471167154, −2.96496618499959719124294818757, −2.52360650851270657096482031713, −1.41831744440605824089144794619, 0.796272406823512472722899510459, 2.19029414507194780538078788873, 3.60521666296040553981642966656, 4.13232637488911848024089390187, 5.07240785226806622648452244836, 5.77113891218550676512444141671, 6.62489756948638298719950280601, 7.19042346445716640881808066150, 7.69742705579854761416431309255, 8.567794357204104931228172467641

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