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Properties

Label 2-525-5.4-c5-0-70
Degree $2$
Conductor $525$
Sign $0.894 + 0.447i$
Analytic cond. $84.2015$
Root an. cond. $9.17613$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.12i·2-s − 9i·3-s − 18.6·4-s + 64.0·6-s − 49i·7-s + 94.7i·8-s − 81·9-s + 643.·11-s + 168. i·12-s − 483. i·13-s + 348.·14-s − 1.27e3·16-s − 295. i·17-s − 576. i·18-s + 1.44e3·19-s + ⋯
L(s)  = 1  + 1.25i·2-s − 0.577i·3-s − 0.584·4-s + 0.726·6-s − 0.377i·7-s + 0.523i·8-s − 0.333·9-s + 1.60·11-s + 0.337i·12-s − 0.792i·13-s + 0.475·14-s − 1.24·16-s − 0.247i·17-s − 0.419i·18-s + 0.921·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(84.2015\)
Root analytic conductor: \(9.17613\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.903434944\)
\(L(\frac12)\) \(\approx\) \(1.903434944\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9iT \)
5 \( 1 \)
7 \( 1 + 49iT \)
good2 \( 1 - 7.12iT - 32T^{2} \)
11 \( 1 - 643.T + 1.61e5T^{2} \)
13 \( 1 + 483. iT - 3.71e5T^{2} \)
17 \( 1 + 295. iT - 1.41e6T^{2} \)
19 \( 1 - 1.44e3T + 2.47e6T^{2} \)
23 \( 1 + 3.92e3iT - 6.43e6T^{2} \)
29 \( 1 + 7.47e3T + 2.05e7T^{2} \)
31 \( 1 + 860.T + 2.86e7T^{2} \)
37 \( 1 - 404. iT - 6.93e7T^{2} \)
41 \( 1 + 1.10e4T + 1.15e8T^{2} \)
43 \( 1 - 1.00e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.54e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.49e4iT - 4.18e8T^{2} \)
59 \( 1 + 4.49e4T + 7.14e8T^{2} \)
61 \( 1 - 3.89e4T + 8.44e8T^{2} \)
67 \( 1 + 1.34e3iT - 1.35e9T^{2} \)
71 \( 1 - 3.15e4T + 1.80e9T^{2} \)
73 \( 1 + 1.05e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.00e4T + 3.07e9T^{2} \)
83 \( 1 + 8.35e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.01e5T + 5.58e9T^{2} \)
97 \( 1 + 1.45e5iT - 8.58e9T^{2} \)
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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

−9.710974222294738375020497419544, −8.808168588440831639167944232892, −7.964155681765758281942551566835, −7.16414489388355283779547670890, −6.51251225270192078097042671739, −5.70500205684218473682810295665, −4.60688863295104965343594024288, −3.27103778898836846435011653182, −1.74659510180977945415552404320, −0.44312141595758585985318560560, 1.20689984924819612232525805766, 2.05384119257252761984516064690, 3.52811046609489298067791774796, 3.88402586538890262325552122689, 5.23752328478689177466282777918, 6.41155189078421613955763601491, 7.42807840298459353012407177064, 9.160301155364239334813129575386, 9.193193231861236866850203708771, 10.15961508608852901000910072160

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