Properties

Label 2-475-475.111-c1-0-31
Degree $2$
Conductor $475$
Sign $-0.977 - 0.209i$
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  + (−1.41 − 0.0992i)2-s + (−0.0334 − 0.956i)3-s + (0.0247 + 0.00348i)4-s + (−0.415 + 2.19i)5-s + (−0.0475 + 1.36i)6-s + (−2.33 − 4.04i)7-s + (2.74 + 0.584i)8-s + (2.07 − 0.145i)9-s + (0.808 − 3.07i)10-s + (0.0421 + 0.401i)11-s + (0.00250 − 0.0238i)12-s + (−0.895 − 1.32i)13-s + (2.91 + 5.97i)14-s + (2.11 + 0.324i)15-s + (−3.89 − 1.11i)16-s + (−2.86 + 7.08i)17-s + ⋯
L(s)  = 1  + (−1.00 − 0.0701i)2-s + (−0.0192 − 0.552i)3-s + (0.0123 + 0.00174i)4-s + (−0.185 + 0.982i)5-s + (−0.0194 + 0.555i)6-s + (−0.882 − 1.52i)7-s + (0.971 + 0.206i)8-s + (0.692 − 0.0484i)9-s + (0.255 − 0.973i)10-s + (0.0127 + 0.120i)11-s + (0.000722 − 0.00687i)12-s + (−0.248 − 0.368i)13-s + (0.778 + 1.59i)14-s + (0.546 + 0.0837i)15-s + (−0.973 − 0.279i)16-s + (−0.694 + 1.71i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.977 - 0.209i$
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: \((2,\ 475,\ (\ :1/2),\ -0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0129389 + 0.122052i\)
\(L(\frac12)\) \(\approx\) \(0.0129389 + 0.122052i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.415 - 2.19i)T \)
19 \( 1 + (-1.82 + 3.95i)T \)
good2 \( 1 + (1.41 + 0.0992i)T + (1.98 + 0.278i)T^{2} \)
3 \( 1 + (0.0334 + 0.956i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (2.33 + 4.04i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0421 - 0.401i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (0.895 + 1.32i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (2.86 - 7.08i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (6.35 + 6.13i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (-2.06 - 5.12i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (6.84 - 7.60i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (2.88 + 2.09i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (9.32 + 2.67i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (0.874 - 4.95i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.811 + 2.00i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (-0.317 - 0.0445i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (-0.937 - 3.75i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (5.29 + 5.10i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (1.58 + 0.987i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (5.70 + 3.03i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (-0.795 + 1.17i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (0.123 + 3.52i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (-2.01 + 2.23i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (0.720 - 0.206i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (-6.72 + 4.20i)T + (42.5 - 87.1i)T^{2} \)
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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

−10.47124212124334267060301556544, −9.940949440165159071520727565279, −8.685577949171319509466667034957, −7.68096206139453563124690265442, −6.99924308908401178396519580517, −6.48518060992889804552464847200, −4.47447723502632139747515858071, −3.51579673825132402729638700021, −1.72615268472012246753703125933, −0.10477952304249649950302426558, 1.92976561793681446065745402061, 3.75801761543425727886515306059, 4.86115111626309232672166290884, 5.76560997647604876452721949734, 7.20957677721503040335883918053, 8.163518781705007014914551780518, 9.113953683372406725808480882712, 9.555405187026173212058189289126, 9.959274456603581603899832945777, 11.58185599260914604051581880999

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