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Properties

Label 2-2475-99.43-c0-0-0
Degree $2$
Conductor $2475$
Sign $0.173 - 0.984i$
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  + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)23-s − 0.999·27-s + (0.5 + 0.866i)31-s − 0.999·33-s + 0.999·36-s + 37-s + 0.999·44-s + (−0.5 + 0.866i)47-s − 0.999·48-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)23-s − 0.999·27-s + (0.5 + 0.866i)31-s − 0.999·33-s + 0.999·36-s + 37-s + 0.999·44-s + (−0.5 + 0.866i)47-s − 0.999·48-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1726, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096223793\)
\(L(\frac12)\) \(\approx\) \(1.096223793\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.410985929848712615106790997803, −8.758412589084946704168195073635, −7.88052780456012402018296595116, −7.07865715415286509771607565017, −5.94360258333511489418202447152, −5.13858760138922981260008018134, −4.68314943332594180782099277072, −3.73230311150702403781378349087, −2.70152505316010835054888682479, −1.52901858271369773888075768574, 0.72137328655715355456477280561, 2.43856473416329474941547616601, 3.00898745401341516618974520873, 3.96925233145204346311140761045, 4.92364133138918527906150437846, 6.01732056864407023036729448778, 6.77191794842081015367782634723, 7.59522228250637489045989516451, 8.223371281512659741714185689608, 8.701952470505225732289211597338

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