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Properties

Label 2-2475-2475.439-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 + (−0.773 + 1.73i)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 + (−1.01 + 1.40i)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 + (−0.773 + 1.73i)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 + (−1.01 + 1.40i)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: $\chi_{2475} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.889 - 0.457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8966158683\)
\(L(\frac12)\) \(\approx\) \(0.8966158683\)
\(L(1)\) 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 \( 1 + (0.773 - 1.73i)T + (-0.669 - 0.743i)T^{2} \)
29 \( 1 + (0.104 - 0.994i)T^{2} \)
31 \( 1 + (-0.139 + 0.155i)T + (-0.104 - 0.994i)T^{2} \)
37 \( 1 + (1.01 - 1.40i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.978 + 0.207i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-1.28 + 1.15i)T + (0.104 - 0.994i)T^{2} \)
53 \( 1 + (-1.89 - 0.614i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.190 - 1.81i)T + (-0.978 - 0.207i)T^{2} \)
61 \( 1 + (0.978 - 0.207i)T^{2} \)
67 \( 1 + (0.309 + 0.278i)T + (0.104 + 0.994i)T^{2} \)
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 \( 1 + (-1.47 + 1.33i)T + (0.104 - 0.994i)T^{2} \)
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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

−9.223228950087743274116703222020, −8.052404325763406108132028910174, −7.32571138091651582195799529803, −7.11508836812394664107607675763, −6.12799668667375182439052158040, −5.39493143139728706823329575434, −4.48211213347641354490367770801, −3.53666090554972045261018804167, −2.25436646942267717905438957629, −1.25299889490315737941194821845, 0.74346940783691358153538109713, 2.39651316418630201557158528871, 3.56912365754733297973592704748, 4.09465494477764304259871434329, 5.22557595446662179729070905449, 6.03728695072949077159912951262, 6.72616275964928241778160178760, 7.36126531808088166294324695570, 8.289048470408066311154838525467, 8.881185273732547820836808839680

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