Properties

Label 2-2475-1.1-c1-0-65
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  + 0.414·2-s − 1.82·4-s + 4.82·7-s − 1.58·8-s + 11-s − 5.65·13-s + 1.99·14-s + 3·16-s − 6.82·17-s − 1.17·19-s + 0.414·22-s − 4·23-s − 2.34·26-s − 8.82·28-s − 0.828·29-s + 4.41·32-s − 2.82·34-s − 0.343·37-s − 0.485·38-s + 0.828·41-s + 3.17·43-s − 1.82·44-s − 1.65·46-s − 4·47-s + 16.3·49-s + 10.3·52-s − 13.3·53-s + ⋯
L(s)  = 1  + 0.292·2-s − 0.914·4-s + 1.82·7-s − 0.560·8-s + 0.301·11-s − 1.56·13-s + 0.534·14-s + 0.750·16-s − 1.65·17-s − 0.268·19-s + 0.0883·22-s − 0.834·23-s − 0.459·26-s − 1.66·28-s − 0.153·29-s + 0.780·32-s − 0.485·34-s − 0.0564·37-s − 0.0787·38-s + 0.129·41-s + 0.483·43-s − 0.275·44-s − 0.244·46-s − 0.583·47-s + 2.33·49-s + 1.43·52-s − 1.82·53-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 \( 1 - 0.414T + 2T^{2} \)
7 \( 1 - 4.82T + 7T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + 6.82T + 17T^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 0.828T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 0.343T + 37T^{2} \)
41 \( 1 - 0.828T + 41T^{2} \)
43 \( 1 - 3.17T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 0.343T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 - 7.65T + 89T^{2} \)
97 \( 1 + 0.343T + 97T^{2} \)
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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

−8.513919456537368504398299507875, −7.916052298769149406857064929715, −7.14448861378128696754356355404, −6.04352277637619096256287229548, −5.07494275294566641492761933675, −4.61189264607412807046133003673, −4.07211395306853007804500071477, −2.57677189011361346570474515760, −1.64464510029904934324792656499, 0, 1.64464510029904934324792656499, 2.57677189011361346570474515760, 4.07211395306853007804500071477, 4.61189264607412807046133003673, 5.07494275294566641492761933675, 6.04352277637619096256287229548, 7.14448861378128696754356355404, 7.916052298769149406857064929715, 8.513919456537368504398299507875

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