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Properties

Label 2-475-95.48-c1-0-19
Degree $2$
Conductor $475$
Sign $0.191 + 0.981i$
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  + (1.06 − 1.51i)2-s + (−0.101 + 1.16i)3-s + (−0.485 − 1.33i)4-s + (1.65 + 1.38i)6-s + (−1.17 − 4.38i)7-s + (1.03 + 0.278i)8-s + (1.60 + 0.283i)9-s + (−0.761 − 1.31i)11-s + (1.60 − 0.429i)12-s + (−0.138 + 0.0120i)13-s + (−7.89 − 2.87i)14-s + (3.69 − 3.10i)16-s + (4.15 + 2.90i)17-s + (2.13 − 2.13i)18-s + (4.14 − 1.34i)19-s + ⋯
L(s)  = 1  + (0.749 − 1.07i)2-s + (−0.0588 + 0.672i)3-s + (−0.242 − 0.666i)4-s + (0.675 + 0.567i)6-s + (−0.444 − 1.65i)7-s + (0.367 + 0.0984i)8-s + (0.536 + 0.0945i)9-s + (−0.229 − 0.397i)11-s + (0.462 − 0.123i)12-s + (−0.0383 + 0.00335i)13-s + (−2.10 − 0.767i)14-s + (0.924 − 0.775i)16-s + (1.00 + 0.705i)17-s + (0.503 − 0.503i)18-s + (0.951 − 0.308i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.191 + 0.981i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.191 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65159 - 1.36084i\)
\(L(\frac12)\) \(\approx\) \(1.65159 - 1.36084i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.14 + 1.34i)T \)
good2 \( 1 + (-1.06 + 1.51i)T + (-0.684 - 1.87i)T^{2} \)
3 \( 1 + (0.101 - 1.16i)T + (-2.95 - 0.520i)T^{2} \)
7 \( 1 + (1.17 + 4.38i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.761 + 1.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.138 - 0.0120i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (-4.15 - 2.90i)T + (5.81 + 15.9i)T^{2} \)
23 \( 1 + (6.70 + 3.12i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-0.346 + 1.96i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.08 + 0.625i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.05 - 2.05i)T + 37iT^{2} \)
41 \( 1 + (-1.05 - 1.25i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-0.373 - 0.801i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (1.41 + 2.02i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (4.74 - 10.1i)T + (-34.0 - 40.6i)T^{2} \)
59 \( 1 + (-1.86 - 10.5i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.79 - 1.01i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (11.6 - 8.15i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (1.79 - 4.92i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-6.01 - 0.526i)T + (71.8 + 12.6i)T^{2} \)
79 \( 1 + (4.43 - 3.72i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.458 - 0.122i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.43 + 1.20i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.682 + 0.975i)T + (-33.1 - 91.1i)T^{2} \)
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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

−10.64252249406452673436077444245, −10.27670157975841622693137355989, −9.657035574865475862106969819298, −7.954270684064504153904063154003, −7.23504674547033379388813195350, −5.78944599824718821858754708738, −4.48170068503732683910609761319, −3.96861614735552807351975415621, −3.06600443414393836467214244164, −1.24148537014966289026729782209, 1.86101051623185919341845475041, 3.41324098748610291668381956042, 4.94673278747172186484274533212, 5.71416399849822497329147640766, 6.40268436847482992096497815198, 7.44483082698505874940138266648, 8.006373483505719225117260481823, 9.392290324500688921656819391348, 10.05548169679271898427999310138, 11.68429917358950212809287468448

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