Properties

Label 2-2475-1.1-c1-0-70
Degree $2$
Conductor $2475$
Sign $-1$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s + 0.999·4-s − 2·7-s − 1.73·8-s + 11-s + 1.46·13-s − 3.46·14-s − 5·16-s − 1.46·19-s + 1.73·22-s − 6.92·23-s + 2.53·26-s − 1.99·28-s − 3.46·29-s + 2.92·31-s − 5.19·32-s − 8.92·37-s − 2.53·38-s + 3.46·41-s − 8.92·43-s + 0.999·44-s − 11.9·46-s + 6.92·47-s − 3·49-s + 1.46·52-s − 12.9·53-s + 3.46·56-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.499·4-s − 0.755·7-s − 0.612·8-s + 0.301·11-s + 0.406·13-s − 0.925·14-s − 1.25·16-s − 0.335·19-s + 0.369·22-s − 1.44·23-s + 0.497·26-s − 0.377·28-s − 0.643·29-s + 0.525·31-s − 0.918·32-s − 1.46·37-s − 0.411·38-s + 0.541·41-s − 1.36·43-s + 0.150·44-s − 1.76·46-s + 1.01·47-s − 0.428·49-s + 0.203·52-s − 1.77·53-s + 0.462·56-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good2 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
13 \( 1 - 1.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 1.46T + 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 - 2.92T + 31T^{2} \)
37 \( 1 + 8.92T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 + 8.92T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.607522049627000704940417341322, −7.66149334077370576666142539244, −6.54973818543039877891372524390, −6.21411664904222640178475119308, −5.37114300791228383583758529819, −4.46344441284911895297876364495, −3.71454119464702276352815475694, −3.07618491095699059708805535565, −1.88100372135999436373448821297, 0, 1.88100372135999436373448821297, 3.07618491095699059708805535565, 3.71454119464702276352815475694, 4.46344441284911895297876364495, 5.37114300791228383583758529819, 6.21411664904222640178475119308, 6.54973818543039877891372524390, 7.66149334077370576666142539244, 8.607522049627000704940417341322

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