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Properties

Label 1-2205-2205.13-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.594 - 0.804i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (−0.997 − 0.0747i)2-s + (0.988 + 0.149i)4-s + (−0.974 − 0.222i)8-s + (0.0747 − 0.997i)11-s + (0.563 − 0.826i)13-s + (0.955 + 0.294i)16-s + (−0.781 + 0.623i)17-s + 19-s + (−0.149 + 0.988i)22-s + (0.149 − 0.988i)23-s + (−0.623 + 0.781i)26-s + (0.988 − 0.149i)29-s + (0.5 + 0.866i)31-s + (−0.930 − 0.365i)32-s + (0.826 − 0.563i)34-s + ⋯
L(s)  = 1  + (−0.997 − 0.0747i)2-s + (0.988 + 0.149i)4-s + (−0.974 − 0.222i)8-s + (0.0747 − 0.997i)11-s + (0.563 − 0.826i)13-s + (0.955 + 0.294i)16-s + (−0.781 + 0.623i)17-s + 19-s + (−0.149 + 0.988i)22-s + (0.149 − 0.988i)23-s + (−0.623 + 0.781i)26-s + (0.988 − 0.149i)29-s + (0.5 + 0.866i)31-s + (−0.930 − 0.365i)32-s + (0.826 − 0.563i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.594 - 0.804i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.594 - 0.804i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9519249746 - 0.4803745501i\)
\(L(\frac12)\) \(\approx\) \(0.9519249746 - 0.4803745501i\)
\(L(1)\) \(\approx\) \(0.7544839584 - 0.1275317907i\)
\(L(1)\) \(\approx\) \(0.7544839584 - 0.1275317907i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.997 - 0.0747i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (0.563 - 0.826i)T \)
17 \( 1 + (-0.781 + 0.623i)T \)
19 \( 1 + T \)
23 \( 1 + (0.149 - 0.988i)T \)
29 \( 1 + (0.988 - 0.149i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.781 - 0.623i)T \)
41 \( 1 + (-0.955 + 0.294i)T \)
43 \( 1 + (-0.294 + 0.955i)T \)
47 \( 1 + (0.997 + 0.0747i)T \)
53 \( 1 + (0.781 + 0.623i)T \)
59 \( 1 + (-0.733 + 0.680i)T \)
61 \( 1 + (0.988 - 0.149i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (-0.433 + 0.900i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.563 - 0.826i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.87548236834703207580347234636, −18.98668987406192676309340235397, −18.33284661276233751410131258459, −17.75982594219190346753553604027, −17.071986914545652354136478582118, −16.32160129198285762431329986522, −15.53523619250263755287965395371, −15.17583707871833639067462808458, −14.01210548199295727038474818674, −13.42623485703908246062694586612, −12.19725051780771462129234937671, −11.68735060068276414009810858627, −11.0497816073555201314841483421, −10.01140301741501745824651554797, −9.54814739354596678082627076648, −8.822635901325519138640211437707, −8.00424693511327761932935983385, −7.0869344479623451350440825480, −6.73188569318760894807149915441, −5.67313569065170782017318334098, −4.74511243275691031789410583085, −3.707695143901835356056751646164, −2.63476586654755178508277277135, −1.84896799089472998951655680294, −0.93170025539440243475892225440, 0.64508812461601443976758226951, 1.40338424413875280973439870068, 2.69551401044577560303190810897, 3.19148685439955370314763207221, 4.32037096554052987134674771464, 5.5821819264693898716894685157, 6.22441761348440220917897670795, 6.97105529568863273269573674565, 7.99752515152982363273029741301, 8.501668921234822742153682847468, 9.11531697345842309651757341764, 10.19914212532586713827571601543, 10.65189593605185958941171713783, 11.38917531297603039158486813858, 12.13485465053983334481043939828, 13.00189351292017420706255052648, 13.79467767263750239180818212816, 14.73570557124699174304632700118, 15.58036027450425039427499782653, 16.09228502691531881788321068586, 16.78201977082819862880605033444, 17.61803639828441916980865470933, 18.1866651495227225190805784486, 18.77985006970218680247763897026, 19.71263771884806308388252031493

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