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Properties

Label 2-475-475.111-c1-0-24
Degree $2$
Conductor $475$
Sign $0.0440 - 0.999i$
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  + (2.44 + 0.170i)2-s + (0.0837 + 2.39i)3-s + (3.96 + 0.557i)4-s + (−2.23 − 0.0696i)5-s + (−0.205 + 5.87i)6-s + (0.533 + 0.923i)7-s + (4.79 + 1.01i)8-s + (−2.74 + 0.192i)9-s + (−5.45 − 0.552i)10-s + (−0.231 − 2.20i)11-s + (−1.00 + 9.54i)12-s + (2.78 + 4.12i)13-s + (1.14 + 2.34i)14-s + (−0.0202 − 5.36i)15-s + (3.85 + 1.10i)16-s + (1.29 − 3.21i)17-s + ⋯
L(s)  = 1  + (1.72 + 0.120i)2-s + (0.0483 + 1.38i)3-s + (1.98 + 0.278i)4-s + (−0.999 − 0.0311i)5-s + (−0.0837 + 2.39i)6-s + (0.201 + 0.348i)7-s + (1.69 + 0.360i)8-s + (−0.915 + 0.0640i)9-s + (−1.72 − 0.174i)10-s + (−0.0697 − 0.663i)11-s + (−0.289 + 2.75i)12-s + (0.772 + 1.14i)13-s + (0.306 + 0.627i)14-s + (−0.00521 − 1.38i)15-s + (0.964 + 0.276i)16-s + (0.315 − 0.779i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0440 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0440 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.0440 - 0.999i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.0440 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38872 + 2.28578i\)
\(L(\frac12)\) \(\approx\) \(2.38872 + 2.28578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.23 + 0.0696i)T \)
19 \( 1 + (3.65 + 2.37i)T \)
good2 \( 1 + (-2.44 - 0.170i)T + (1.98 + 0.278i)T^{2} \)
3 \( 1 + (-0.0837 - 2.39i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (-0.533 - 0.923i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.231 + 2.20i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (-2.78 - 4.12i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (-1.29 + 3.21i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (-3.33 - 3.21i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (3.33 + 8.25i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (-3.80 + 4.22i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (0.811 + 0.589i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.99 - 1.43i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (-0.548 + 3.11i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.403 - 0.997i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (10.8 + 1.52i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (-1.92 - 7.71i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (-3.32 - 3.21i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (-5.64 - 3.52i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (6.08 + 3.23i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (5.55 - 8.23i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (0.0538 + 1.54i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (-9.40 + 10.4i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (8.36 - 2.39i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (10.6 - 6.68i)T + (42.5 - 87.1i)T^{2} \)
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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

−11.39972938774959157107281240702, −10.88928210497383014920296284013, −9.459195614558302622676447479809, −8.577653884417340387005265189908, −7.30522403755164063048484581391, −6.17676050660132828297532404110, −5.18395142289899975418774239278, −4.29293442223546261022586995009, −3.84420168735021630587280790734, −2.75315268605439239987946232896, 1.43330212572091923691164373831, 2.91081694195319862646502066352, 3.90069312062401584510733542232, 4.96589203906100137837321243653, 6.15677220092985563184762966830, 6.89148025165218626076544288054, 7.71311892246594383217331910275, 8.462113197939917748837731285369, 10.61160534850874713201533140132, 11.06713245726600359184347503904

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