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Properties

Label 2-475-95.18-c1-0-22
Degree $2$
Conductor $475$
Sign $0.993 + 0.116i$
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  + (1.10 + 1.10i)2-s + (1.29 − 1.29i)3-s + 0.460i·4-s + 2.87·6-s + (1.53 − 1.53i)7-s + (1.70 − 1.70i)8-s − 0.369i·9-s − 2.63·11-s + (0.598 + 0.598i)12-s + (−1.29 + 1.29i)13-s + 3.41·14-s + 4.70·16-s + (0.709 − 0.709i)17-s + (0.409 − 0.409i)18-s + (−3.41 − 2.70i)19-s + ⋯
L(s)  = 1  + (0.784 + 0.784i)2-s + (0.749 − 0.749i)3-s + 0.230i·4-s + 1.17·6-s + (0.581 − 0.581i)7-s + (0.603 − 0.603i)8-s − 0.123i·9-s − 0.793·11-s + (0.172 + 0.172i)12-s + (−0.359 + 0.359i)13-s + 0.912·14-s + 1.17·16-s + (0.172 − 0.172i)17-s + (0.0965 − 0.0965i)18-s + (−0.783 − 0.621i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.993 + 0.116i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.72694 - 0.159908i\)
\(L(\frac12)\) \(\approx\) \(2.72694 - 0.159908i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.41 + 2.70i)T \)
good2 \( 1 + (-1.10 - 1.10i)T + 2iT^{2} \)
3 \( 1 + (-1.29 + 1.29i)T - 3iT^{2} \)
7 \( 1 + (-1.53 + 1.53i)T - 7iT^{2} \)
11 \( 1 + 2.63T + 11T^{2} \)
13 \( 1 + (1.29 - 1.29i)T - 13iT^{2} \)
17 \( 1 + (-0.709 + 0.709i)T - 17iT^{2} \)
23 \( 1 + (-1.53 - 1.53i)T + 23iT^{2} \)
29 \( 1 - 1.84T + 29T^{2} \)
31 \( 1 - 10.8iT - 31T^{2} \)
37 \( 1 + (-3.51 - 3.51i)T + 37iT^{2} \)
41 \( 1 - 5.83iT - 41T^{2} \)
43 \( 1 + (8.21 + 8.21i)T + 43iT^{2} \)
47 \( 1 + (-6.80 + 6.80i)T - 47iT^{2} \)
53 \( 1 + (6.11 - 6.11i)T - 53iT^{2} \)
59 \( 1 + 5.83T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + (1.67 + 1.67i)T + 67iT^{2} \)
71 \( 1 + 3.99iT - 71T^{2} \)
73 \( 1 + (-7.38 - 7.38i)T + 73iT^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + (2.51 + 2.51i)T + 83iT^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + (9.14 + 9.14i)T + 97iT^{2} \)
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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

−10.89429513187502397729071560710, −10.23002064539000822637834456257, −8.860829548259079007498997653515, −7.922990613287865187770798506618, −7.26275367758690065171100110815, −6.56298928922587295245610410652, −5.17552257414054684480666150113, −4.53083989763404805026689428356, −3.00483126178853947453695930712, −1.53913190337415048040514718727, 2.18101855294755992936398226964, 3.01888638299714304047381301464, 4.10960107903681521350772473122, 4.88598716456205911111101137420, 5.97578176038577993268015739327, 7.79009726934809165033639456357, 8.291242396687459928845471209764, 9.378511903227281424731341707709, 10.31663217204052553637342490409, 11.03112084428825922857872026802

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