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Properties

Label 2-2475-2475.439-c0-0-1
Degree $2$
Conductor $2475$
Sign $0.361 + 0.932i$
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  + (0.669 − 0.743i)3-s + (0.978 − 0.207i)4-s + (0.5 − 0.866i)5-s + (−0.104 − 0.994i)9-s + (0.104 + 0.994i)11-s + (0.5 − 0.866i)12-s + (−0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (0.309 − 0.951i)20-s + (−0.773 + 1.73i)23-s + (−0.499 − 0.866i)25-s + (−0.809 − 0.587i)27-s + (−1.22 + 1.35i)31-s + (0.809 + 0.587i)33-s + (−0.309 − 0.951i)36-s + (1.01 − 1.40i)37-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)3-s + (0.978 − 0.207i)4-s + (0.5 − 0.866i)5-s + (−0.104 − 0.994i)9-s + (0.104 + 0.994i)11-s + (0.5 − 0.866i)12-s + (−0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (0.309 − 0.951i)20-s + (−0.773 + 1.73i)23-s + (−0.499 − 0.866i)25-s + (−0.809 − 0.587i)27-s + (−1.22 + 1.35i)31-s + (0.809 + 0.587i)33-s + (−0.309 − 0.951i)36-s + (1.01 − 1.40i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.361 + 0.932i$
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: \((2,\ 2475,\ (\ :0),\ 0.361 + 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.029587901\)
\(L(\frac12)\) \(\approx\) \(2.029587901\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 + 0.743i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
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 \( 1 + (1.22 - 1.35i)T + (-0.104 - 0.994i)T^{2} \)
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 \( 1 + (0.773 + 0.251i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.0218 + 0.207i)T + (-0.978 - 0.207i)T^{2} \)
61 \( 1 + (0.978 - 0.207i)T^{2} \)
67 \( 1 + (1.10 + 0.994i)T + (0.104 + 0.994i)T^{2} \)
71 \( 1 + (0.0646 - 0.198i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (0.604 - 0.544i)T + (0.104 - 0.994i)T^{2} \)
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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

−9.196567247000995957883389504134, −7.890566876406639146212460800148, −7.61115156494476540747834618549, −6.72328474430142422870557954869, −5.97712733089365775626424148846, −5.23822837513038096960687400287, −4.03634419515177311112924142442, −2.97871737748097036557225297174, −1.88093963889910835817637242934, −1.47437297861495226244911632841, 1.93170421524188985969780546660, 2.70034019215075739913234938285, 3.36537491146888314161254079254, 4.24246943933933485696406497530, 5.53897276012751695739434970795, 6.20882215154697909134114045697, 6.91369312994526680051507379066, 7.88356327531066514672829405313, 8.374478799391406424472458081658, 9.335892685467624237712828089248

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