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Properties

Label 2-475-95.9-c1-0-14
Degree $2$
Conductor $475$
Sign $0.690 + 0.723i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
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  + (0.207 − 0.0366i)2-s + (0.0513 + 0.0612i)3-s + (−1.83 + 0.668i)4-s + (0.0129 + 0.0108i)6-s + (−1.46 − 0.843i)7-s + (−0.722 + 0.417i)8-s + (0.519 − 2.94i)9-s + (1.44 + 2.50i)11-s + (−0.135 − 0.0781i)12-s + (4.15 − 4.95i)13-s + (−0.334 − 0.121i)14-s + (2.86 − 2.40i)16-s + (2.94 − 0.518i)17-s − 0.631i·18-s + (4.34 + 0.288i)19-s + ⋯
L(s)  = 1  + (0.146 − 0.0259i)2-s + (0.0296 + 0.0353i)3-s + (−0.918 + 0.334i)4-s + (0.00527 + 0.00442i)6-s + (−0.552 − 0.318i)7-s + (−0.255 + 0.147i)8-s + (0.173 − 0.982i)9-s + (0.435 + 0.753i)11-s + (−0.0390 − 0.0225i)12-s + (1.15 − 1.37i)13-s + (−0.0894 − 0.0325i)14-s + (0.715 − 0.600i)16-s + (0.713 − 0.125i)17-s − 0.148i·18-s + (0.997 + 0.0662i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09587 - 0.468860i\)
\(L(\frac12)\) \(\approx\) \(1.09587 - 0.468860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (-4.34 - 0.288i)T \)
good2 \( 1 + (-0.207 + 0.0366i)T + (1.87 - 0.684i)T^{2} \)
3 \( 1 + (-0.0513 - 0.0612i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (1.46 + 0.843i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.44 - 2.50i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.15 + 4.95i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.94 + 0.518i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (2.82 + 7.75i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.26 - 7.14i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.02 - 3.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.96iT - 37T^{2} \)
41 \( 1 + (-4.17 + 3.50i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.82 - 5.01i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.62 + 0.286i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (0.653 + 1.79i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.616 + 3.49i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-7.42 + 2.70i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.23 + 0.393i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (9.79 + 3.56i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-0.907 - 1.08i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-1.84 + 1.54i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-9.87 - 5.70i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.74 - 6.50i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (9.71 - 1.71i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.71937863597470124920793028048, −9.888360619025494373721503994900, −9.170412560719356231241941382589, −8.296170649059984899592157730893, −7.26077256174845185878592707789, −6.16729072437128932852685307012, −5.12708427990186652693867087276, −3.81750952036051144636443135351, −3.27463738631818672229260286754, −0.829472107878822969016180300753, 1.46687042565100987532776733918, 3.38883399142834457518741858281, 4.29637582747518277242130319374, 5.58638502554721261867332594656, 6.17155639838315313665585758174, 7.60402570247491512438199903757, 8.539789199411964785766210477801, 9.415873754378035998431021199737, 9.973364939768050762462928951489, 11.27170951834501698215901881037

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