Properties

Label 2-2475-2475.2221-c0-0-1
Degree $2$
Conductor $2475$
Sign $0.335 - 0.942i$
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  + (−0.978 + 0.207i)3-s + (−0.104 + 0.994i)4-s + 5-s + (0.913 − 0.406i)9-s + (0.669 + 0.743i)11-s + (−0.104 − 0.994i)12-s + (−0.978 + 0.207i)15-s + (−0.978 − 0.207i)16-s + (−0.104 + 0.994i)20-s + (1.58 − 0.336i)23-s + 25-s + (−0.809 + 0.587i)27-s + (1.22 + 0.544i)31-s + (−0.809 − 0.587i)33-s + (0.309 + 0.951i)36-s + (−0.309 − 0.951i)37-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)3-s + (−0.104 + 0.994i)4-s + 5-s + (0.913 − 0.406i)9-s + (0.669 + 0.743i)11-s + (−0.104 − 0.994i)12-s + (−0.978 + 0.207i)15-s + (−0.978 − 0.207i)16-s + (−0.104 + 0.994i)20-s + (1.58 − 0.336i)23-s + 25-s + (−0.809 + 0.587i)27-s + (1.22 + 0.544i)31-s + (−0.809 − 0.587i)33-s + (0.309 + 0.951i)36-s + (−0.309 − 0.951i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.335 - 0.942i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (2221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.335 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096682686\)
\(L(\frac12)\) \(\approx\) \(1.096682686\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.978 - 0.207i)T \)
5 \( 1 - T \)
11 \( 1 + (-0.669 - 0.743i)T \)
good2 \( 1 + (0.104 - 0.994i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.104 + 0.994i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-1.58 + 0.336i)T + (0.913 - 0.406i)T^{2} \)
29 \( 1 + (-0.669 + 0.743i)T^{2} \)
31 \( 1 + (-1.22 - 0.544i)T + (0.669 + 0.743i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.104 + 0.994i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \)
53 \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (1.30 - 1.45i)T + (-0.104 - 0.994i)T^{2} \)
61 \( 1 + (0.104 - 0.994i)T^{2} \)
67 \( 1 + (0.190 + 0.0850i)T + (0.669 + 0.743i)T^{2} \)
71 \( 1 + (-1.58 - 1.14i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.669 + 0.743i)T^{2} \)
83 \( 1 + (0.978 - 0.207i)T^{2} \)
89 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-1.22 + 0.544i)T + (0.669 - 0.743i)T^{2} \)
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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

−9.320045632180466271794135256883, −8.681417830652494103742712136169, −7.56313566188084008622679144304, −6.72612804276870582011530672927, −6.42045663193914660849473363845, −5.15564184878782810373884244864, −4.70947308103997580972938868409, −3.69959529415523795914345385408, −2.61537426290941559899913345331, −1.36851558810799209782976405018, 0.988861015885058307228252231399, 1.74890988081040408612731150279, 3.11693920674292507162562823943, 4.67038331740424333419290591717, 5.05799800669720770065754605978, 6.10367518175023266605675135746, 6.27926744969575668108267556061, 7.06908796902213714945377607382, 8.290984839840925573675480257987, 9.314740180116530819737189763647

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