Properties

Label 2-3525-1.1-c1-0-19
Degree $2$
Conductor $3525$
Sign $1$
Analytic cond. $28.1472$
Root an. cond. $5.30539$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.267·2-s − 3-s − 1.92·4-s + 0.267·6-s + 0.207·7-s + 1.04·8-s + 9-s − 1.27·11-s + 1.92·12-s + 5.37·13-s − 0.0553·14-s + 3.57·16-s − 0.969·17-s − 0.267·18-s + 6.42·19-s − 0.207·21-s + 0.340·22-s − 5.87·23-s − 1.04·24-s − 1.43·26-s − 27-s − 0.399·28-s − 9.93·29-s + 5.97·31-s − 3.05·32-s + 1.27·33-s + 0.259·34-s + ⋯
L(s)  = 1  − 0.188·2-s − 0.577·3-s − 0.964·4-s + 0.109·6-s + 0.0783·7-s + 0.371·8-s + 0.333·9-s − 0.383·11-s + 0.556·12-s + 1.49·13-s − 0.0147·14-s + 0.894·16-s − 0.235·17-s − 0.0629·18-s + 1.47·19-s − 0.0452·21-s + 0.0725·22-s − 1.22·23-s − 0.214·24-s − 0.281·26-s − 0.192·27-s − 0.0755·28-s − 1.84·29-s + 1.07·31-s − 0.540·32-s + 0.221·33-s + 0.0444·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $1$
Analytic conductor: \(28.1472\)
Root analytic conductor: \(5.30539\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.024806062\)
\(L(\frac12)\) \(\approx\) \(1.024806062\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
47 \( 1 - T \)
good2 \( 1 + 0.267T + 2T^{2} \)
7 \( 1 - 0.207T + 7T^{2} \)
11 \( 1 + 1.27T + 11T^{2} \)
13 \( 1 - 5.37T + 13T^{2} \)
17 \( 1 + 0.969T + 17T^{2} \)
19 \( 1 - 6.42T + 19T^{2} \)
23 \( 1 + 5.87T + 23T^{2} \)
29 \( 1 + 9.93T + 29T^{2} \)
31 \( 1 - 5.97T + 31T^{2} \)
37 \( 1 - 1.19T + 37T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 - 6.20T + 43T^{2} \)
53 \( 1 - 8.64T + 53T^{2} \)
59 \( 1 + 9.70T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 4.36T + 67T^{2} \)
71 \( 1 - 9.54T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + 1.79T + 83T^{2} \)
89 \( 1 + 7.28T + 89T^{2} \)
97 \( 1 - 0.290T + 97T^{2} \)
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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

−8.553036737516490207407932762422, −7.895496322336472721776251450537, −7.25539080592416205237553956847, −5.96673716698341771912699336611, −5.75229077171789423552272465412, −4.74472427067016655893150762445, −4.01553204214416328496478119368, −3.24018962172131346501451702935, −1.68948806147203965253899989810, −0.66734418222847149311091553215, 0.66734418222847149311091553215, 1.68948806147203965253899989810, 3.24018962172131346501451702935, 4.01553204214416328496478119368, 4.74472427067016655893150762445, 5.75229077171789423552272465412, 5.96673716698341771912699336611, 7.25539080592416205237553956847, 7.895496322336472721776251450537, 8.553036737516490207407932762422

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