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Properties

Label 2-525-105.59-c1-0-13
Degree $2$
Conductor $525$
Sign $-0.0331 - 0.999i$
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  + (1.12 + 1.94i)2-s + (−1.42 − 0.983i)3-s + (−1.53 + 2.65i)4-s + (0.313 − 3.88i)6-s + (2.22 − 1.42i)7-s − 2.39·8-s + (1.06 + 2.80i)9-s + (1.64 + 0.952i)11-s + (4.79 − 2.27i)12-s + 5.07·13-s + (5.29 + 2.73i)14-s + (0.369 + 0.639i)16-s + (−3.85 − 2.22i)17-s + (−4.26 + 5.22i)18-s + (−3.85 + 2.22i)19-s + ⋯
L(s)  = 1  + (0.795 + 1.37i)2-s + (−0.822 − 0.568i)3-s + (−0.766 + 1.32i)4-s + (0.128 − 1.58i)6-s + (0.841 − 0.540i)7-s − 0.846·8-s + (0.354 + 0.935i)9-s + (0.497 + 0.287i)11-s + (1.38 − 0.656i)12-s + 1.40·13-s + (1.41 + 0.730i)14-s + (0.0923 + 0.159i)16-s + (−0.936 − 0.540i)17-s + (−1.00 + 1.23i)18-s + (−0.884 + 0.510i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0331 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0331 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0331 - 0.999i$
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: \((2,\ 525,\ (\ :1/2),\ -0.0331 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30778 + 1.35193i\)
\(L(\frac12)\) \(\approx\) \(1.30778 + 1.35193i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.42 + 0.983i)T \)
5 \( 1 \)
7 \( 1 + (-2.22 + 1.42i)T \)
good2 \( 1 + (-1.12 - 1.94i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-1.64 - 0.952i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.07T + 13T^{2} \)
17 \( 1 + (3.85 + 2.22i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.85 - 2.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.42 - 2.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.82iT - 29T^{2} \)
31 \( 1 + (-4.81 - 2.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.02 + 2.32i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.250T + 41T^{2} \)
43 \( 1 + 9.23iT - 43T^{2} \)
47 \( 1 + (-2.66 + 1.53i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.21 - 2.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.95 + 12.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.51 - 0.874i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.18 + 4.14i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.68iT - 71T^{2} \)
73 \( 1 + (3.15 - 5.47i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.59 + 2.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.98iT - 83T^{2} \)
89 \( 1 + (5.43 + 9.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.94T + 97T^{2} \)
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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

−11.10917878695092547146114361245, −10.57866413328732981503791324047, −8.810592969106067094951109761169, −8.067586131736856250446422733804, −7.05542203743497520387647776789, −6.59683552171736455081174664405, −5.62045584556323101272554644586, −4.75147349174016751333149470119, −3.91480172760618025605593603029, −1.52572751290983187104046179814, 1.19243264171681211115593944472, 2.62382359642543863197363719238, 4.17491790147644893076756451338, 4.41311486771783887150610284884, 5.75258020298631389875069851263, 6.41002304884139913317645187471, 8.292518101170389864516600452450, 9.151496648986098673142041457312, 10.21990331883291840094101618150, 11.09078452029506729401058386937

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