Properties

Label 2-2475-1.1-c1-0-54
Degree $2$
Conductor $2475$
Sign $-1$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·7-s + 3·8-s − 11-s − 2·13-s − 3·14-s − 16-s − 3·17-s − 19-s + 22-s − 23-s + 2·26-s − 3·28-s − 6·29-s + 4·31-s − 5·32-s + 3·34-s − 37-s + 38-s + 5·41-s − 4·43-s + 44-s + 46-s − 3·47-s + 2·49-s + 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.13·7-s + 1.06·8-s − 0.301·11-s − 0.554·13-s − 0.801·14-s − 1/4·16-s − 0.727·17-s − 0.229·19-s + 0.213·22-s − 0.208·23-s + 0.392·26-s − 0.566·28-s − 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.514·34-s − 0.164·37-s + 0.162·38-s + 0.780·41-s − 0.609·43-s + 0.150·44-s + 0.147·46-s − 0.437·47-s + 2/7·49-s + 0.277·52-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: yes
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 \( 1 + T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−8.503195786802655471275015261860, −7.925108494003391656542258459331, −7.37326409570226547636119611448, −6.31018569651781389923413435679, −5.13569250388043882655281554123, −4.72076358349491808912876066304, −3.80888706948583729809072577283, −2.35667797071707152789075717627, −1.42734846445239921757802359874, 0, 1.42734846445239921757802359874, 2.35667797071707152789075717627, 3.80888706948583729809072577283, 4.72076358349491808912876066304, 5.13569250388043882655281554123, 6.31018569651781389923413435679, 7.37326409570226547636119611448, 7.925108494003391656542258459331, 8.503195786802655471275015261860

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