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Properties

Label 2-525-21.20-c1-0-37
Degree $2$
Conductor $525$
Sign $0.198 + 0.980i$
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  − 1.09i·2-s + (1.70 + 0.323i)3-s + 0.791·4-s + (0.355 − 1.87i)6-s + (−1 − 2.44i)7-s − 3.06i·8-s + (2.79 + 1.09i)9-s − 3.06i·11-s + (1.34 + 0.255i)12-s + 2.44i·13-s + (−2.69 + 1.09i)14-s − 1.79·16-s − 2.69·17-s + (1.20 − 3.06i)18-s + 4.38i·19-s + ⋯
L(s)  = 1  − 0.777i·2-s + (0.982 + 0.186i)3-s + 0.395·4-s + (0.144 − 0.763i)6-s + (−0.377 − 0.925i)7-s − 1.08i·8-s + (0.930 + 0.366i)9-s − 0.925i·11-s + (0.388 + 0.0737i)12-s + 0.679i·13-s + (−0.719 + 0.293i)14-s − 0.447·16-s − 0.653·17-s + (0.284 − 0.723i)18-s + 1.00i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.198 + 0.980i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71889 - 1.40543i\)
\(L(\frac12)\) \(\approx\) \(1.71889 - 1.40543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 0.323i)T \)
5 \( 1 \)
7 \( 1 + (1 + 2.44i)T \)
good2 \( 1 + 1.09iT - 2T^{2} \)
11 \( 1 + 3.06iT - 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + 2.69T + 17T^{2} \)
19 \( 1 - 4.38iT - 19T^{2} \)
23 \( 1 + 5.26iT - 23T^{2} \)
29 \( 1 - 5.26iT - 29T^{2} \)
31 \( 1 - 6.83iT - 31T^{2} \)
37 \( 1 - 8.58T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + 6.58T + 43T^{2} \)
47 \( 1 + 2.69T + 47T^{2} \)
53 \( 1 - 3.93iT - 53T^{2} \)
59 \( 1 + 7.51T + 59T^{2} \)
61 \( 1 + 6.83iT - 61T^{2} \)
67 \( 1 - 4.16T + 67T^{2} \)
71 \( 1 + 3.06iT - 71T^{2} \)
73 \( 1 - 16.1iT - 73T^{2} \)
79 \( 1 - 0.582T + 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 - 7.51T + 89T^{2} \)
97 \( 1 + 11.7iT - 97T^{2} \)
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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

−10.66301043152650757542992734677, −9.938068612402823623396829819780, −9.063267957841084733808083192382, −8.093638043177411917726600995931, −7.06273485932057114445505261655, −6.32235190768800284100971946711, −4.43829051129235516338085071396, −3.60286749513328941185883069297, −2.70710607130775556586788946663, −1.34192785938245072143428077102, 2.11770585924029337569141189331, 2.88293043121573528269811973287, 4.47115881451899639836627766342, 5.72999566426945757149500549630, 6.62330686110557125292468298426, 7.53318446133461467571291601985, 8.139105203807625809435300692093, 9.211427605692900067295535871429, 9.764921638772443644393603630508, 11.12921407975172304860370337638

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