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Properties

Label 2-475-475.6-c1-0-0
Degree $2$
Conductor $475$
Sign $-0.368 - 0.929i$
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  + (−1.91 − 1.19i)2-s + (−2.99 + 0.857i)3-s + (1.35 + 2.78i)4-s + (0.364 − 2.20i)5-s + (6.74 + 1.93i)6-s + (−1.35 + 2.35i)7-s + (0.257 − 2.45i)8-s + (5.66 − 3.54i)9-s + (−3.33 + 3.78i)10-s + (−0.206 + 0.228i)11-s + (−6.44 − 7.15i)12-s + (−0.225 − 6.45i)13-s + (5.41 − 2.87i)14-s + (0.802 + 6.91i)15-s + (0.380 − 0.486i)16-s + (1.39 − 0.0978i)17-s + ⋯
L(s)  = 1  + (−1.35 − 0.845i)2-s + (−1.72 + 0.495i)3-s + (0.678 + 1.39i)4-s + (0.162 − 0.986i)5-s + (2.75 + 0.790i)6-s + (−0.513 + 0.888i)7-s + (0.0911 − 0.867i)8-s + (1.88 − 1.18i)9-s + (−1.05 + 1.19i)10-s + (−0.0621 + 0.0690i)11-s + (−1.85 − 2.06i)12-s + (−0.0625 − 1.79i)13-s + (1.44 − 0.768i)14-s + (0.207 + 1.78i)15-s + (0.0950 − 0.121i)16-s + (0.339 − 0.0237i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.368 - 0.929i$
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: \((2,\ 475,\ (\ :1/2),\ -0.368 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00659840 + 0.00971705i\)
\(L(\frac12)\) \(\approx\) \(0.00659840 + 0.00971705i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.364 + 2.20i)T \)
19 \( 1 + (3.90 + 1.93i)T \)
good2 \( 1 + (1.91 + 1.19i)T + (0.876 + 1.79i)T^{2} \)
3 \( 1 + (2.99 - 0.857i)T + (2.54 - 1.58i)T^{2} \)
7 \( 1 + (1.35 - 2.35i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.206 - 0.228i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (0.225 + 6.45i)T + (-12.9 + 0.906i)T^{2} \)
17 \( 1 + (-1.39 + 0.0978i)T + (16.8 - 2.36i)T^{2} \)
23 \( 1 + (-6.04 + 0.849i)T + (22.1 - 6.33i)T^{2} \)
29 \( 1 + (-6.50 - 0.454i)T + (28.7 + 4.03i)T^{2} \)
31 \( 1 + (7.52 - 3.35i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (2.30 - 7.09i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.87 - 3.67i)T + (-9.91 - 39.7i)T^{2} \)
43 \( 1 + (-0.355 - 2.01i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (2.84 + 0.198i)T + (46.5 + 6.54i)T^{2} \)
53 \( 1 + (-0.0893 - 0.183i)T + (-32.6 + 41.7i)T^{2} \)
59 \( 1 + (2.93 + 7.25i)T + (-42.4 + 40.9i)T^{2} \)
61 \( 1 + (8.94 - 1.25i)T + (58.6 - 16.8i)T^{2} \)
67 \( 1 + (3.07 + 12.3i)T + (-59.1 + 31.4i)T^{2} \)
71 \( 1 + (-1.75 - 1.69i)T + (2.47 + 70.9i)T^{2} \)
73 \( 1 + (0.237 - 6.81i)T + (-72.8 - 5.09i)T^{2} \)
79 \( 1 + (6.62 - 1.89i)T + (66.9 - 41.8i)T^{2} \)
83 \( 1 + (0.932 - 0.415i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-1.24 - 1.59i)T + (-21.5 + 86.3i)T^{2} \)
97 \( 1 + (2.54 - 10.2i)T + (-85.6 - 45.5i)T^{2} \)
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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

−10.99307839123645591677010636537, −10.45374271880065521897840495467, −9.705142297168358139577246539147, −8.944785866681828932672318181193, −8.043399116963646316238099554913, −6.58512615101648065837796393157, −5.49389256091423410196602049510, −4.90365732150876818484323669520, −3.04201461941784314203523833254, −1.16884953373656207572817025405, 0.01656011009592763422604138043, 1.61799139912954196292619656306, 4.11441895806578939081807228480, 5.68282585166389854595815166660, 6.52386139399621684619386063298, 6.97598319095033263238571730189, 7.47150944576504841322682438214, 9.018449568249917945976622574311, 10.01710642426877536263886472097, 10.64628086728832855873053876797

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