Properties

Label 2-3525-1.1-c1-0-29
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  − 0.237·2-s − 3-s − 1.94·4-s + 0.237·6-s − 1.64·7-s + 0.935·8-s + 9-s + 5.84·11-s + 1.94·12-s + 4.61·13-s + 0.390·14-s + 3.66·16-s + 5.67·17-s − 0.237·18-s − 6.71·19-s + 1.64·21-s − 1.38·22-s + 6.07·23-s − 0.935·24-s − 1.09·26-s − 27-s + 3.20·28-s + 3.23·29-s − 6.81·31-s − 2.74·32-s − 5.84·33-s − 1.34·34-s + ⋯
L(s)  = 1  − 0.167·2-s − 0.577·3-s − 0.971·4-s + 0.0968·6-s − 0.622·7-s + 0.330·8-s + 0.333·9-s + 1.76·11-s + 0.561·12-s + 1.27·13-s + 0.104·14-s + 0.916·16-s + 1.37·17-s − 0.0559·18-s − 1.54·19-s + 0.359·21-s − 0.295·22-s + 1.26·23-s − 0.190·24-s − 0.214·26-s − 0.192·27-s + 0.604·28-s + 0.600·29-s − 1.22·31-s − 0.484·32-s − 1.01·33-s − 0.230·34-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

\(L(1)\) \(\approx\) \(1.199410484\)
\(L(\frac12)\) \(\approx\) \(1.199410484\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.237T + 2T^{2} \)
7 \( 1 + 1.64T + 7T^{2} \)
11 \( 1 - 5.84T + 11T^{2} \)
13 \( 1 - 4.61T + 13T^{2} \)
17 \( 1 - 5.67T + 17T^{2} \)
19 \( 1 + 6.71T + 19T^{2} \)
23 \( 1 - 6.07T + 23T^{2} \)
29 \( 1 - 3.23T + 29T^{2} \)
31 \( 1 + 6.81T + 31T^{2} \)
37 \( 1 + 5.84T + 37T^{2} \)
41 \( 1 + 2.60T + 41T^{2} \)
43 \( 1 + 0.899T + 43T^{2} \)
53 \( 1 + 4.11T + 53T^{2} \)
59 \( 1 - 6.93T + 59T^{2} \)
61 \( 1 + 6.54T + 61T^{2} \)
67 \( 1 - 9.16T + 67T^{2} \)
71 \( 1 - 3.98T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + 7.03T + 89T^{2} \)
97 \( 1 + 8.59T + 97T^{2} \)
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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

−8.730384685123854153448636822809, −8.012986812495194300646577243194, −6.80891908289831514000523886241, −6.43128559024784776152745787218, −5.58475608559302111917758045707, −4.77327371634645298372999038530, −3.71185550161639886041529030545, −3.54764114881775072239638008476, −1.58798079713987883517378571123, −0.75194083378277500445155014235, 0.75194083378277500445155014235, 1.58798079713987883517378571123, 3.54764114881775072239638008476, 3.71185550161639886041529030545, 4.77327371634645298372999038530, 5.58475608559302111917758045707, 6.43128559024784776152745787218, 6.80891908289831514000523886241, 8.012986812495194300646577243194, 8.730384685123854153448636822809

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