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Properties

Label 2-2475-2475.736-c0-0-0
Degree $2$
Conductor $2475$
Sign $0.889 - 0.457i$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
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  + (0.913 + 0.406i)3-s + (−0.978 + 0.207i)4-s + 5-s + (0.669 + 0.743i)9-s + (−0.104 − 0.994i)11-s + (−0.978 − 0.207i)12-s + (0.913 + 0.406i)15-s + (0.913 − 0.406i)16-s + (−0.978 + 0.207i)20-s + (0.564 + 0.251i)23-s + 25-s + (0.309 + 0.951i)27-s + (−0.139 + 0.155i)31-s + (0.309 − 0.951i)33-s + (−0.809 − 0.587i)36-s + (0.809 + 0.587i)37-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)3-s + (−0.978 + 0.207i)4-s + 5-s + (0.669 + 0.743i)9-s + (−0.104 − 0.994i)11-s + (−0.978 − 0.207i)12-s + (0.913 + 0.406i)15-s + (0.913 − 0.406i)16-s + (−0.978 + 0.207i)20-s + (0.564 + 0.251i)23-s + 25-s + (0.309 + 0.951i)27-s + (−0.139 + 0.155i)31-s + (0.309 − 0.951i)33-s + (−0.809 − 0.587i)36-s + (0.809 + 0.587i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.889 - 0.457i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (736, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.889 - 0.457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.625539288\)
\(L(\frac12)\) \(\approx\) \(1.625539288\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 - T \)
11 \( 1 + (0.104 + 0.994i)T \)
good2 \( 1 + (0.978 - 0.207i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.978 + 0.207i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.564 - 0.251i)T + (0.669 + 0.743i)T^{2} \)
29 \( 1 + (0.104 - 0.994i)T^{2} \)
31 \( 1 + (0.139 - 0.155i)T + (-0.104 - 0.994i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.978 + 0.207i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \)
53 \( 1 + (0.0646 - 0.198i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.190 - 1.81i)T + (-0.978 - 0.207i)T^{2} \)
61 \( 1 + (0.978 - 0.207i)T^{2} \)
67 \( 1 + (1.30 - 1.45i)T + (-0.104 - 0.994i)T^{2} \)
71 \( 1 + (-0.564 + 1.73i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (-0.913 - 0.406i)T^{2} \)
89 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.139 + 0.155i)T + (-0.104 + 0.994i)T^{2} \)
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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

−9.099293336265856392960973584141, −8.604348789288167083423780529714, −7.955491310995363225044575060047, −6.97216476382835437047665983740, −5.86940594840458558404168175306, −5.16775908672463786607882698014, −4.37249436498973704271429644883, −3.38391879688236996125148813364, −2.72223649762545417619980977721, −1.36645771942227172053390946162, 1.28077797525219540513342890289, 2.23480330638013447112340848134, 3.23612908992767590342830814018, 4.34245637835334034758709577462, 4.99221113520090816164621909598, 6.00584344821403110131752318424, 6.77589698498494023028058901683, 7.67891427742275835711538307619, 8.372368068572486634289680487522, 9.290221175437424524295385734646

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