Properties

Label 2-475-19.9-c1-0-8
Degree $2$
Conductor $475$
Sign $-0.902 - 0.429i$
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  + (0.340 + 1.93i)2-s + (0.143 − 0.120i)3-s + (−1.73 + 0.632i)4-s + (0.282 + 0.236i)6-s + (−0.338 + 0.586i)7-s + (0.148 + 0.257i)8-s + (−0.514 + 2.91i)9-s + (−1.42 − 2.46i)11-s + (−0.173 + 0.300i)12-s + (3.65 + 3.06i)13-s + (−1.24 − 0.454i)14-s + (−3.27 + 2.75i)16-s + (0.900 + 5.10i)17-s − 5.81·18-s + (−3.00 − 3.15i)19-s + ⋯
L(s)  = 1  + (0.240 + 1.36i)2-s + (0.0830 − 0.0696i)3-s + (−0.868 + 0.316i)4-s + (0.115 + 0.0966i)6-s + (−0.127 + 0.221i)7-s + (0.0525 + 0.0909i)8-s + (−0.171 + 0.973i)9-s + (−0.428 − 0.741i)11-s + (−0.0500 + 0.0867i)12-s + (1.01 + 0.849i)13-s + (−0.333 − 0.121i)14-s + (−0.819 + 0.687i)16-s + (0.218 + 1.23i)17-s − 1.37·18-s + (−0.690 − 0.723i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.902 - 0.429i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.902 - 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.336975 + 1.49203i\)
\(L(\frac12)\) \(\approx\) \(0.336975 + 1.49203i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.00 + 3.15i)T \)
good2 \( 1 + (-0.340 - 1.93i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (-0.143 + 0.120i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.338 - 0.586i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.42 + 2.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.65 - 3.06i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.900 - 5.10i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (0.987 - 0.359i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.247 - 1.40i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (0.135 - 0.234i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.603T + 37T^{2} \)
41 \( 1 + (-5.15 + 4.32i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-5.28 - 1.92i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.37 + 7.77i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (6.47 - 2.35i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.75 + 9.96i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-7.02 + 2.55i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.714 + 4.05i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-7.14 - 2.60i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-12.3 + 10.3i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-11.3 + 9.54i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (7.09 - 12.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.31 - 5.29i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.994 + 5.64i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.99373121220382537345902352917, −10.85395935530187430508180184615, −9.150163188959327375641819914483, −8.379799689673197394333439800081, −7.80841284930373666293453910656, −6.61130592255681706977810899929, −5.96614394436805904127684458261, −5.04542496500548501168000535794, −3.87787821113013498877404134777, −2.13609963183231893677830249642, 0.892975052571346723149517760604, 2.50125846359962675657183639814, 3.50357631152469365037351145003, 4.38244891760268481182079131582, 5.74360806784802789774054300618, 6.92060313208369574143498526012, 8.047510141396268667872540009238, 9.265106245447154079604329273635, 9.926872888539035248153394937787, 10.70944258272753121482698305298

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