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Properties

Label 2-525-105.59-c1-0-1
Degree $2$
Conductor $525$
Sign $0.154 - 0.988i$
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  + (−0.846 − 1.46i)2-s + (−1.53 + 0.803i)3-s + (−0.433 + 0.750i)4-s + (2.47 + 1.57i)6-s + (2.01 + 1.71i)7-s − 1.91·8-s + (1.71 − 2.46i)9-s + (−0.399 − 0.230i)11-s + (0.0622 − 1.49i)12-s − 3.38·13-s + (0.803 − 4.40i)14-s + (2.49 + 4.31i)16-s + (−4.76 − 2.75i)17-s + (−5.06 − 0.421i)18-s + (−3.49 + 2.01i)19-s + ⋯
L(s)  = 1  + (−0.598 − 1.03i)2-s + (−0.886 + 0.463i)3-s + (−0.216 + 0.375i)4-s + (1.01 + 0.641i)6-s + (0.762 + 0.647i)7-s − 0.678·8-s + (0.570 − 0.821i)9-s + (−0.120 − 0.0695i)11-s + (0.0179 − 0.432i)12-s − 0.938·13-s + (0.214 − 1.17i)14-s + (0.622 + 1.07i)16-s + (−1.15 − 0.667i)17-s + (−1.19 − 0.0992i)18-s + (−0.801 + 0.462i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.154 - 0.988i$
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: \((2,\ 525,\ (\ :1/2),\ 0.154 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.221307 + 0.189404i\)
\(L(\frac12)\) \(\approx\) \(0.221307 + 0.189404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.53 - 0.803i)T \)
5 \( 1 \)
7 \( 1 + (-2.01 - 1.71i)T \)
good2 \( 1 + (0.846 + 1.46i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (0.399 + 0.230i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.38T + 13T^{2} \)
17 \( 1 + (4.76 + 2.75i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.49 - 2.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.25 - 3.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.71iT - 29T^{2} \)
31 \( 1 + (3.01 + 1.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.02 + 2.89i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.25T + 41T^{2} \)
43 \( 1 - 8.35iT - 43T^{2} \)
47 \( 1 + (-2.73 + 1.57i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.78 - 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.88 - 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.90 - 3.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.793 - 0.458i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.52iT - 71T^{2} \)
73 \( 1 + (-3.89 + 6.75i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.58 + 6.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.5iT - 83T^{2} \)
89 \( 1 + (-1.35 - 2.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.44T + 97T^{2} \)
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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

−10.96350978692872869946265858949, −10.47665486355961150396432989188, −9.318010098569044392952077139088, −8.985104297334593873800521483261, −7.55850016414623521930591037310, −6.32662488832756785127222035416, −5.34012494213675595621075057495, −4.42737099885947188116812877193, −2.87781293681056351803310174520, −1.60484191899186015898076957605, 0.22542626831394471628631374930, 2.18979156653017079658117105181, 4.35175149603051981703930991924, 5.22937807647876707540200365190, 6.47357974295338220314473781155, 6.91344349611808536365387497481, 7.86785653418402098412417741750, 8.481546955184556596412731125956, 9.723289641815233044472560801919, 10.71712985492619850031485990324

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