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Properties

Label 2-3620-3620.2899-c0-0-1
Degree $2$
Conductor $3620$
Sign $-0.972 - 0.231i$
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  + (−0.559 − 0.829i)2-s + (0.442 + 1.77i)3-s + (−0.374 + 0.927i)4-s + (−0.104 + 0.994i)5-s + (1.22 − 1.35i)6-s + (0.173 − 0.300i)7-s + (0.978 − 0.207i)8-s + (−2.06 + 1.09i)9-s + (0.882 − 0.469i)10-s + (−1.80 − 0.254i)12-s + (−0.346 + 0.0242i)14-s + (−1.80 + 0.254i)15-s + (−0.719 − 0.694i)16-s + (2.06 + 1.09i)18-s + (−0.882 − 0.469i)20-s + (0.609 + 0.174i)21-s + ⋯
L(s)  = 1  + (−0.559 − 0.829i)2-s + (0.442 + 1.77i)3-s + (−0.374 + 0.927i)4-s + (−0.104 + 0.994i)5-s + (1.22 − 1.35i)6-s + (0.173 − 0.300i)7-s + (0.978 − 0.207i)8-s + (−2.06 + 1.09i)9-s + (0.882 − 0.469i)10-s + (−1.80 − 0.254i)12-s + (−0.346 + 0.0242i)14-s + (−1.80 + 0.254i)15-s + (−0.719 − 0.694i)16-s + (2.06 + 1.09i)18-s + (−0.882 − 0.469i)20-s + (0.609 + 0.174i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3620\)    =    \(2^{2} \cdot 5 \cdot 181\)
Sign: $-0.972 - 0.231i$
Analytic conductor: \(1.80661\)
Root analytic conductor: \(1.34410\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3620} (2899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3620,\ (\ :0),\ -0.972 - 0.231i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7105711348\)
\(L(\frac12)\) \(\approx\) \(0.7105711348\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.559 + 0.829i)T \)
5 \( 1 + (0.104 - 0.994i)T \)
181 \( 1 + (0.374 - 0.927i)T \)
good3 \( 1 + (-0.442 - 1.77i)T + (-0.882 + 0.469i)T^{2} \)
7 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.241 - 0.970i)T^{2} \)
13 \( 1 + (-0.990 - 0.139i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.0168 - 0.483i)T + (-0.997 + 0.0697i)T^{2} \)
29 \( 1 + (1.88 - 0.399i)T + (0.913 - 0.406i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.848 - 0.529i)T^{2} \)
41 \( 1 + (1.59 - 0.995i)T + (0.438 - 0.898i)T^{2} \)
43 \( 1 + (0.671 + 0.563i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-1.93 - 0.272i)T + (0.961 + 0.275i)T^{2} \)
53 \( 1 + (-0.438 - 0.898i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.59 - 0.580i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.139 + 0.155i)T + (-0.104 - 0.994i)T^{2} \)
71 \( 1 + (-0.669 - 0.743i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.438 - 0.898i)T^{2} \)
83 \( 1 + (0.270 - 0.555i)T + (-0.615 - 0.788i)T^{2} \)
89 \( 1 + (-0.704 - 0.256i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (0.241 + 0.970i)T^{2} \)
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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

−9.222264972683962026163088894930, −8.686997966797136971910632279412, −7.76934304025951735605972673307, −7.22029219494851860442063990283, −5.90215897921252399340440195444, −4.95440794130549597961638499734, −4.12909963835446505052546025565, −3.52807240682813753227941662631, −2.97502216614446239990769276769, −1.95965385181022934707670014218, 0.45416868822729459147581761814, 1.59649002387813853163787727308, 2.19129185759996494754353552626, 3.69867201054688984344819134611, 4.94192743663819806491819660527, 5.69886227222190916992993017679, 6.26440211028953419554287085598, 7.20024217508486264229187611566, 7.59649332374091933460187370991, 8.310307414699653166168222073820

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