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Properties

Label 2-525-5.3-c2-0-3
Degree $2$
Conductor $525$
Sign $-0.850 + 0.525i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.408 + 0.408i)2-s + (1.22 + 1.22i)3-s + 3.66i·4-s − 1.00·6-s + (1.87 − 1.87i)7-s + (−3.13 − 3.13i)8-s + 2.99i·9-s − 6.25·11-s + (−4.49 + 4.49i)12-s + (−16.4 − 16.4i)13-s + 1.52i·14-s − 12.1·16-s + (−20.4 + 20.4i)17-s + (−1.22 − 1.22i)18-s + 7.15i·19-s + ⋯
L(s)  = 1  + (−0.204 + 0.204i)2-s + (0.408 + 0.408i)3-s + 0.916i·4-s − 0.166·6-s + (0.267 − 0.267i)7-s + (−0.391 − 0.391i)8-s + 0.333i·9-s − 0.568·11-s + (−0.374 + 0.374i)12-s + (−1.26 − 1.26i)13-s + 0.109i·14-s − 0.756·16-s + (−1.20 + 1.20i)17-s + (−0.0680 − 0.0680i)18-s + 0.376i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4272013986\)
\(L(\frac12)\) \(\approx\) \(0.4272013986\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
7 \( 1 + (-1.87 + 1.87i)T \)
good2 \( 1 + (0.408 - 0.408i)T - 4iT^{2} \)
11 \( 1 + 6.25T + 121T^{2} \)
13 \( 1 + (16.4 + 16.4i)T + 169iT^{2} \)
17 \( 1 + (20.4 - 20.4i)T - 289iT^{2} \)
19 \( 1 - 7.15iT - 361T^{2} \)
23 \( 1 + (-12.0 - 12.0i)T + 529iT^{2} \)
29 \( 1 + 18.1iT - 841T^{2} \)
31 \( 1 + 33.3T + 961T^{2} \)
37 \( 1 + (18.8 - 18.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 50.8T + 1.68e3T^{2} \)
43 \( 1 + (53.3 + 53.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (46.9 - 46.9i)T - 2.20e3iT^{2} \)
53 \( 1 + (-28.9 - 28.9i)T + 2.80e3iT^{2} \)
59 \( 1 - 10.0iT - 3.48e3T^{2} \)
61 \( 1 - 85.6T + 3.72e3T^{2} \)
67 \( 1 + (11.9 - 11.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 20.8T + 5.04e3T^{2} \)
73 \( 1 + (35.2 + 35.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 31.9iT - 6.24e3T^{2} \)
83 \( 1 + (6.49 + 6.49i)T + 6.88e3iT^{2} \)
89 \( 1 - 145. iT - 7.92e3T^{2} \)
97 \( 1 + (31.5 - 31.5i)T - 9.40e3iT^{2} \)
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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

−10.98908758404783626750405668120, −10.26026765250800141935669811475, −9.300060980185625160913529343811, −8.327311919428784018891734798639, −7.79102644581404628833076808513, −6.93396379293393291957774771533, −5.50091742488291532182855705706, −4.40865020848928433624121529264, −3.38217120271940339070791566923, −2.29917463834557028663220210097, 0.15466891763095184089500950522, 1.89493845080922070068061106488, 2.66001627432618605473387243335, 4.59300747273624371615592415910, 5.27846363961274411271090824350, 6.70912728939115290724754639864, 7.20715536731676562638435665055, 8.631227966399135723019173438161, 9.244449249895798286140165358885, 9.966719085837498658820736794835

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