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Properties

Label 2-475-95.18-c1-0-19
Degree $2$
Conductor $475$
Sign $0.525 + 0.850i$
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  − 2i·4-s + (0.679 − 0.679i)7-s + 3i·9-s + 4.35·11-s − 4·16-s + (5.67 − 5.67i)17-s − 4.35i·19-s + (−6.35 − 6.35i)23-s + (−1.35 − 1.35i)28-s + 6·36-s + (7.03 + 7.03i)43-s − 8.71i·44-s + (−4.32 + 4.32i)47-s + 6.07i·49-s + 4.35·61-s + ⋯
L(s)  = 1  − i·4-s + (0.256 − 0.256i)7-s + i·9-s + 1.31·11-s − 16-s + (1.37 − 1.37i)17-s − 0.999i·19-s + (−1.32 − 1.32i)23-s + (−0.256 − 0.256i)28-s + 36-s + (1.07 + 1.07i)43-s − 1.31i·44-s + (−0.630 + 0.630i)47-s + 0.868i·49-s + 0.558·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.525 + 0.850i$
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: \((2,\ 475,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31538 - 0.733375i\)
\(L(\frac12)\) \(\approx\) \(1.31538 - 0.733375i\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.35iT \)
good2 \( 1 + 2iT^{2} \)
3 \( 1 - 3iT^{2} \)
7 \( 1 + (-0.679 + 0.679i)T - 7iT^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-5.67 + 5.67i)T - 17iT^{2} \)
23 \( 1 + (6.35 + 6.35i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-7.03 - 7.03i)T + 43iT^{2} \)
47 \( 1 + (4.32 - 4.32i)T - 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 4.35T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-12.0 - 12.0i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-3.64 - 3.64i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−10.90313932913755348360852228164, −9.915151453094189041748454078275, −9.320657232373119101148748416644, −8.135971961511208869930344976468, −7.11535433642295865400163367946, −6.17013184146683492010185497715, −5.10814855300203410242461842413, −4.29287209788908344502150817662, −2.51759522165962650799529371831, −1.06723956875773738720441433594, 1.68249617262429988445591308594, 3.67502021899078383462489706737, 3.79881751746183031268491832202, 5.67285788013934757179832067609, 6.51591101344694365055661929044, 7.66273146773971366844500387611, 8.389842739316571088399285558315, 9.294981460322697943240898048035, 10.12187420205236215722173167437, 11.48401784080345632185565783914

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