Properties

Label 2-525-1.1-c1-0-4
Degree $2$
Conductor $525$
Sign $1$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
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  − 2.70·2-s + 3-s + 5.34·4-s − 2.70·6-s + 7-s − 9.04·8-s + 9-s + 2·11-s + 5.34·12-s + 0.921·13-s − 2.70·14-s + 13.8·16-s + 1.07·17-s − 2.70·18-s + 3.07·19-s + 21-s − 5.41·22-s + 2.34·23-s − 9.04·24-s − 2.49·26-s + 27-s + 5.34·28-s − 6.68·29-s − 7.75·31-s − 19.3·32-s + 2·33-s − 2.92·34-s + ⋯
L(s)  = 1  − 1.91·2-s + 0.577·3-s + 2.67·4-s − 1.10·6-s + 0.377·7-s − 3.19·8-s + 0.333·9-s + 0.603·11-s + 1.54·12-s + 0.255·13-s − 0.724·14-s + 3.45·16-s + 0.261·17-s − 0.638·18-s + 0.706·19-s + 0.218·21-s − 1.15·22-s + 0.487·23-s − 1.84·24-s − 0.489·26-s + 0.192·27-s + 1.00·28-s − 1.24·29-s − 1.39·31-s − 3.42·32-s + 0.348·33-s − 0.501·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8404594786\)
\(L(\frac12)\) \(\approx\) \(0.8404594786\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good2 \( 1 + 2.70T + 2T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 0.921T + 13T^{2} \)
17 \( 1 - 1.07T + 17T^{2} \)
19 \( 1 - 3.07T + 19T^{2} \)
23 \( 1 - 2.34T + 23T^{2} \)
29 \( 1 + 6.68T + 29T^{2} \)
31 \( 1 + 7.75T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 + 6.52T + 43T^{2} \)
47 \( 1 - 4.68T + 47T^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 + 4.15T + 61T^{2} \)
67 \( 1 - 4.68T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 7.07T + 73T^{2} \)
79 \( 1 - 6.15T + 79T^{2} \)
83 \( 1 - 6.83T + 83T^{2} \)
89 \( 1 - 8.34T + 89T^{2} \)
97 \( 1 + 8.43T + 97T^{2} \)
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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

−10.73093724426907748138347685787, −9.488955697269998902726127343570, −9.311266427577489277333430127999, −8.260314135259506262863829014208, −7.59127026891978329060420902585, −6.82472178482904034539484641598, −5.63385722833276316572021417302, −3.66481207120444030082643463708, −2.33657418047064760612703806245, −1.14951268463002916512455010277, 1.14951268463002916512455010277, 2.33657418047064760612703806245, 3.66481207120444030082643463708, 5.63385722833276316572021417302, 6.82472178482904034539484641598, 7.59127026891978329060420902585, 8.260314135259506262863829014208, 9.311266427577489277333430127999, 9.488955697269998902726127343570, 10.73093724426907748138347685787

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