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Properties

Label 2-525-21.17-c1-0-30
Degree $2$
Conductor $525$
Sign $0.976 + 0.214i$
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  + (−2.01 + 1.16i)2-s + (1.67 + 0.436i)3-s + (1.71 − 2.97i)4-s + (−3.89 + 1.07i)6-s + (−1.11 − 2.39i)7-s + 3.33i·8-s + (2.61 + 1.46i)9-s + (−2.42 − 1.39i)11-s + (4.17 − 4.23i)12-s − 3.20i·13-s + (5.04 + 3.53i)14-s + (−0.459 − 0.795i)16-s + (−0.440 + 0.763i)17-s + (−6.99 + 0.0942i)18-s + (1.90 − 1.09i)19-s + ⋯
L(s)  = 1  + (−1.42 + 0.824i)2-s + (0.967 + 0.252i)3-s + (0.858 − 1.48i)4-s + (−1.58 + 0.437i)6-s + (−0.422 − 0.906i)7-s + 1.18i·8-s + (0.872 + 0.488i)9-s + (−0.729 − 0.421i)11-s + (1.20 − 1.22i)12-s − 0.888i·13-s + (1.34 + 0.946i)14-s + (−0.114 − 0.198i)16-s + (−0.106 + 0.185i)17-s + (−1.64 + 0.0222i)18-s + (0.436 − 0.251i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.976 + 0.214i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.976 + 0.214i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.851602 - 0.0923569i\)
\(L(\frac12)\) \(\approx\) \(0.851602 - 0.0923569i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 - 0.436i)T \)
5 \( 1 \)
7 \( 1 + (1.11 + 2.39i)T \)
good2 \( 1 + (2.01 - 1.16i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (2.42 + 1.39i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.20iT - 13T^{2} \)
17 \( 1 + (0.440 - 0.763i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.90 + 1.09i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.53 + 3.77i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.15iT - 29T^{2} \)
31 \( 1 + (7.62 + 4.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.203 - 0.352i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.55T + 41T^{2} \)
43 \( 1 - 0.118T + 43T^{2} \)
47 \( 1 + (1.31 + 2.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.46 - 3.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.04 - 3.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.7 + 6.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.802 - 1.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.25iT - 71T^{2} \)
73 \( 1 + (0.192 + 0.110i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.56 - 2.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.666T + 83T^{2} \)
89 \( 1 + (-0.437 - 0.757i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.37iT - 97T^{2} \)
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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

−10.47339306482386600903129177615, −9.708168811218493707903286147325, −9.015623746234478365208663429228, −8.052185225602201264861708309053, −7.58743429575727352487405875007, −6.74200795637812113188342343137, −5.49140088021677134348396312664, −3.96486312576730694519264598664, −2.64342590918367373040047214808, −0.75024047564018133650575990499, 1.55407035344953063400655227173, 2.56791214007606880472269628372, 3.43883952528604025299336972170, 5.22577832716884234296456842013, 6.94421195069351172867190564368, 7.53256220945345139442429314962, 8.660373012014567043348202719257, 9.129471741393316304647440993125, 9.690760004050046994498027615955, 10.68837399228100823400369294532

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