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Properties

Label 2-525-105.47-c0-0-1
Degree $2$
Conductor $525$
Sign $0.320 + 0.947i$
Analytic cond. $0.262009$
Root an. cond. $0.511868$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)3-s + (0.866 − 0.5i)4-s + (−0.965 − 0.258i)7-s + (−0.866 − 0.499i)9-s + (−0.258 − 0.965i)12-s + (0.707 + 0.707i)13-s + (0.499 − 0.866i)16-s + (−0.866 + 1.5i)19-s + (−0.499 + 0.866i)21-s + (−0.707 + 0.707i)27-s + (−0.965 + 0.258i)28-s − 36-s + (1.67 − 0.448i)37-s + (0.866 − 0.500i)39-s + (−0.707 − 0.707i)48-s + (0.866 + 0.499i)49-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)3-s + (0.866 − 0.5i)4-s + (−0.965 − 0.258i)7-s + (−0.866 − 0.499i)9-s + (−0.258 − 0.965i)12-s + (0.707 + 0.707i)13-s + (0.499 − 0.866i)16-s + (−0.866 + 1.5i)19-s + (−0.499 + 0.866i)21-s + (−0.707 + 0.707i)27-s + (−0.965 + 0.258i)28-s − 36-s + (1.67 − 0.448i)37-s + (0.866 − 0.500i)39-s + (−0.707 − 0.707i)48-s + (0.866 + 0.499i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.320 + 0.947i$
Analytic conductor: \(0.262009\)
Root analytic conductor: \(0.511868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :0),\ 0.320 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.033658446\)
\(L(\frac12)\) \(\approx\) \(1.033658446\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (0.965 + 0.258i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
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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

−10.97377286728519001795725934686, −10.07191823124196312632056027411, −9.130913236433912530413661621122, −8.041581548739523507988444196374, −7.14490082230783928163982301265, −6.29252985633352818907255064139, −5.88803285784870582058732789287, −3.91490788644184588606427867294, −2.69936208651861485005943151331, −1.48726815656126762725697522187, 2.58214667023386001339226008871, 3.28412892766438771131336611196, 4.42602831652703613530353595059, 5.83022137337127157278780202840, 6.57089840092050160237786398068, 7.77569530867521567904457312425, 8.694022886543941598205848492894, 9.459823793138481294232011613273, 10.52884527146853947743010918930, 11.02543517325953042938133730503

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