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Properties

Label 2-475-475.6-c1-0-5
Degree $2$
Conductor $475$
Sign $-0.167 - 0.985i$
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  + (−0.730 − 0.456i)2-s + (−0.0193 + 0.00554i)3-s + (−0.551 − 1.13i)4-s + (2.18 − 0.484i)5-s + (0.0166 + 0.00477i)6-s + (−1.43 + 2.49i)7-s + (−0.293 + 2.79i)8-s + (−2.54 + 1.58i)9-s + (−1.81 − 0.642i)10-s + (−3.60 + 4.00i)11-s + (0.0169 + 0.0187i)12-s + (−0.125 − 3.59i)13-s + (2.18 − 1.16i)14-s + (−0.0394 + 0.0214i)15-s + (−0.0611 + 0.0782i)16-s + (−5.24 + 0.366i)17-s + ⋯
L(s)  = 1  + (−0.516 − 0.322i)2-s + (−0.0111 + 0.00319i)3-s + (−0.275 − 0.565i)4-s + (0.976 − 0.216i)5-s + (0.00679 + 0.00194i)6-s + (−0.543 + 0.941i)7-s + (−0.103 + 0.986i)8-s + (−0.847 + 0.529i)9-s + (−0.574 − 0.203i)10-s + (−1.08 + 1.20i)11-s + (0.00488 + 0.00542i)12-s + (−0.0348 − 0.997i)13-s + (0.584 − 0.310i)14-s + (−0.0101 + 0.00554i)15-s + (−0.0152 + 0.0195i)16-s + (−1.27 + 0.0889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.167 - 0.985i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.167 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301450 + 0.356885i\)
\(L(\frac12)\) \(\approx\) \(0.301450 + 0.356885i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.18 + 0.484i)T \)
19 \( 1 + (-0.594 - 4.31i)T \)
good2 \( 1 + (0.730 + 0.456i)T + (0.876 + 1.79i)T^{2} \)
3 \( 1 + (0.0193 - 0.00554i)T + (2.54 - 1.58i)T^{2} \)
7 \( 1 + (1.43 - 2.49i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.60 - 4.00i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (0.125 + 3.59i)T + (-12.9 + 0.906i)T^{2} \)
17 \( 1 + (5.24 - 0.366i)T + (16.8 - 2.36i)T^{2} \)
23 \( 1 + (-2.99 + 0.421i)T + (22.1 - 6.33i)T^{2} \)
29 \( 1 + (0.511 + 0.0357i)T + (28.7 + 4.03i)T^{2} \)
31 \( 1 + (5.07 - 2.26i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (1.40 - 4.32i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.686 + 0.878i)T + (-9.91 - 39.7i)T^{2} \)
43 \( 1 + (0.503 + 2.85i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (9.90 + 0.692i)T + (46.5 + 6.54i)T^{2} \)
53 \( 1 + (-0.270 - 0.554i)T + (-32.6 + 41.7i)T^{2} \)
59 \( 1 + (-3.60 - 8.92i)T + (-42.4 + 40.9i)T^{2} \)
61 \( 1 + (0.622 - 0.0874i)T + (58.6 - 16.8i)T^{2} \)
67 \( 1 + (1.23 + 4.95i)T + (-59.1 + 31.4i)T^{2} \)
71 \( 1 + (-9.96 - 9.62i)T + (2.47 + 70.9i)T^{2} \)
73 \( 1 + (-0.267 + 7.66i)T + (-72.8 - 5.09i)T^{2} \)
79 \( 1 + (-6.42 + 1.84i)T + (66.9 - 41.8i)T^{2} \)
83 \( 1 + (-14.9 + 6.65i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-4.87 - 6.24i)T + (-21.5 + 86.3i)T^{2} \)
97 \( 1 + (-2.07 + 8.33i)T + (-85.6 - 45.5i)T^{2} \)
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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

−10.82559439440098300501989552971, −10.34588550226399165664114419432, −9.502705592817003791474074081837, −8.817223596172681396132777145706, −7.962409139466336990855814780484, −6.41182751173649950667197897755, −5.40235541196329804050840771749, −5.06733667268205359103836286795, −2.72384673526658627036271848148, −1.97233324173946830313435727484, 0.31369504299244987815948794064, 2.68479470207005468159287253979, 3.69975346814893735762565182967, 5.14724324138336566979844000195, 6.46550715645239250097271883647, 6.92332541314104297656233637274, 8.172404786019678367451050294612, 9.121499666557087509727659098858, 9.493522896315015837346948021805, 10.80296482712705963303637459113

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