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Properties

Label 2-2475-495.439-c0-0-1
Degree $2$
Conductor $2475$
Sign $0.958 - 0.285i$
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  + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)16-s + (1.73 − i)23-s − 0.999i·27-s + (0.5 + 0.866i)31-s + 0.999i·33-s + 0.999·36-s + i·37-s − 0.999·44-s + (−0.866 − 0.5i)47-s + 0.999i·48-s + (0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)16-s + (1.73 − i)23-s − 0.999i·27-s + (0.5 + 0.866i)31-s + 0.999i·33-s + 0.999·36-s + i·37-s − 0.999·44-s + (−0.866 − 0.5i)47-s + 0.999i·48-s + (0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.958 - 0.285i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1924, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.958 - 0.285i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.819418610\)
\(L(\frac12)\) \(\approx\) \(1.819418610\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.902534164971543007432772578777, −8.307164550381195222140535970065, −7.68746621845476031570971803413, −6.84962999043895390791480594433, −6.59554413612106615183297366653, −5.06357021274888346396416816148, −4.22067478512625066326261983860, −3.12548254049514266161692672447, −2.63632044038703476769391993008, −1.55548732245944899656194330065, 1.27740552360422936359522531804, 2.50666096389941352614683487624, 3.16253534847921224917491391369, 4.27260811534546924976057160885, 5.25116195982091452614199108643, 5.78497627084798025153033917213, 6.90938307792765069434276143368, 7.55790038612016167467163177726, 8.419050657881271565294529317743, 9.198217210896313125337638275079

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