Properties

Label 2-2475-1.1-c1-0-41
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  − 1.30·2-s − 0.302·4-s − 0.697·7-s + 3·8-s + 11-s − 5·13-s + 0.908·14-s − 3.30·16-s + 6.90·17-s − 19-s − 1.30·22-s − 7.30·23-s + 6.51·26-s + 0.211·28-s − 0.908·29-s + 10.2·31-s − 1.69·32-s − 9·34-s − 2.39·37-s + 1.30·38-s + 5.60·41-s − 7.21·43-s − 0.302·44-s + 9.51·46-s − 3·47-s − 6.51·49-s + 1.51·52-s + ⋯
L(s)  = 1  − 0.921·2-s − 0.151·4-s − 0.263·7-s + 1.06·8-s + 0.301·11-s − 1.38·13-s + 0.242·14-s − 0.825·16-s + 1.67·17-s − 0.229·19-s − 0.277·22-s − 1.52·23-s + 1.27·26-s + 0.0398·28-s − 0.168·29-s + 1.83·31-s − 0.300·32-s − 1.54·34-s − 0.393·37-s + 0.211·38-s + 0.875·41-s − 1.09·43-s − 0.0456·44-s + 1.40·46-s − 0.437·47-s − 0.930·49-s + 0.209·52-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 \( 1 + 1.30T + 2T^{2} \)
7 \( 1 + 0.697T + 7T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 - 6.90T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + 7.30T + 23T^{2} \)
29 \( 1 + 0.908T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 2.39T + 37T^{2} \)
41 \( 1 - 5.60T + 41T^{2} \)
43 \( 1 + 7.21T + 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + 1.30T + 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 + 7.90T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 2.60T + 71T^{2} \)
73 \( 1 - 7.90T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 3.51T + 83T^{2} \)
89 \( 1 + 1.69T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−8.415399862659603199416534755798, −7.986651192879807112285336444773, −7.27442983818976872671281423815, −6.38967263571840803143135749724, −5.37353016843940857903175698802, −4.58610882863565380133372931648, −3.65481076245711625675108826063, −2.47159096994729384411981162893, −1.28350279549380974401560284608, 0, 1.28350279549380974401560284608, 2.47159096994729384411981162893, 3.65481076245711625675108826063, 4.58610882863565380133372931648, 5.37353016843940857903175698802, 6.38967263571840803143135749724, 7.27442983818976872671281423815, 7.986651192879807112285336444773, 8.415399862659603199416534755798

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