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Properties

Label 2-525-105.59-c1-0-30
Degree $2$
Conductor $525$
Sign $0.779 + 0.626i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.450 + 0.780i)2-s + (−0.248 − 1.71i)3-s + (0.593 − 1.02i)4-s + (1.22 − 0.966i)6-s + (2.64 − 0.105i)7-s + 2.87·8-s + (−2.87 + 0.853i)9-s + (5.46 + 3.15i)11-s + (−1.91 − 0.761i)12-s − 3.77·13-s + (1.27 + 2.01i)14-s + (0.107 + 0.185i)16-s + (−3.27 − 1.88i)17-s + (−1.96 − 1.86i)18-s + (4.57 − 2.64i)19-s + ⋯
L(s)  = 1  + (0.318 + 0.551i)2-s + (−0.143 − 0.989i)3-s + (0.296 − 0.514i)4-s + (0.500 − 0.394i)6-s + (0.999 − 0.0398i)7-s + 1.01·8-s + (−0.958 + 0.284i)9-s + (1.64 + 0.950i)11-s + (−0.551 − 0.219i)12-s − 1.04·13-s + (0.340 + 0.538i)14-s + (0.0268 + 0.0464i)16-s + (−0.793 − 0.458i)17-s + (−0.462 − 0.438i)18-s + (1.05 − 0.606i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.779 + 0.626i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.779 + 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89694 - 0.667224i\)
\(L(\frac12)\) \(\approx\) \(1.89694 - 0.667224i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.248 + 1.71i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 + 0.105i)T \)
good2 \( 1 + (-0.450 - 0.780i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-5.46 - 3.15i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.77T + 13T^{2} \)
17 \( 1 + (3.27 + 1.88i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.57 + 2.64i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.38 + 5.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.06iT - 29T^{2} \)
31 \( 1 + (-0.349 - 0.201i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.15 - 0.668i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.53T + 41T^{2} \)
43 \( 1 - 4.84iT - 43T^{2} \)
47 \( 1 + (1.68 - 0.970i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.720 - 1.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.60 + 2.77i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.3 - 6.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.68 - 1.55i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.21iT - 71T^{2} \)
73 \( 1 + (1.25 - 2.18i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.05 - 5.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.53iT - 83T^{2} \)
89 \( 1 + (-0.590 - 1.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.10T + 97T^{2} \)
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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

−11.01823264353129534627521413171, −9.857384630974436619496165991888, −8.846906824419141969195636092542, −7.66722133460082783729631680502, −6.99546115802775734213664714085, −6.44065012938480871745058354023, −5.14923530144682856010743629728, −4.50156264376961252224203986014, −2.32547332940166381367706153201, −1.31482885890046875840694785839, 1.79868835254749348647249041864, 3.35087954472886616211966925124, 4.05700275363977090423998247069, 5.02607392621779061734157877134, 6.16858127245750158262354211405, 7.47642881743099329772497246701, 8.407138550640219649107333438030, 9.289275660364529941531320041296, 10.25307789951600404602902403792, 11.19470760747232469597907958939

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