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Properties

Label 2-475-19.11-c1-0-21
Degree $2$
Conductor $475$
Sign $-0.0977 + 0.995i$
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  + (0.610 − 1.05i)2-s + (1.14 − 1.97i)3-s + (0.253 + 0.439i)4-s + (−1.39 − 2.41i)6-s + 1.28·7-s + 3.06·8-s + (−1.11 − 1.92i)9-s + 0.285·11-s + 1.15·12-s + (−2.5 − 4.33i)13-s + (0.785 − 1.35i)14-s + (1.36 − 2.36i)16-s + (−3.11 + 5.40i)17-s − 2.71·18-s + (2.92 + 3.22i)19-s + ⋯
L(s)  = 1  + (0.431 − 0.748i)2-s + (0.659 − 1.14i)3-s + (0.126 + 0.219i)4-s + (−0.569 − 0.987i)6-s + 0.485·7-s + 1.08·8-s + (−0.370 − 0.641i)9-s + 0.0859·11-s + 0.334·12-s + (−0.693 − 1.20i)13-s + (0.209 − 0.363i)14-s + (0.341 − 0.590i)16-s + (−0.756 + 1.30i)17-s − 0.639·18-s + (0.671 + 0.740i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63824 - 1.80701i\)
\(L(\frac12)\) \(\approx\) \(1.63824 - 1.80701i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.92 - 3.22i)T \)
good2 \( 1 + (-0.610 + 1.05i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.14 + 1.97i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.28T + 7T^{2} \)
11 \( 1 - 0.285T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.11 - 5.40i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.61 + 4.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.642 + 1.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.22T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + (-0.420 + 0.728i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.47 - 4.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.86 - 4.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.18 - 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.86 - 4.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.22 + 3.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.492 + 0.853i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.46 + 2.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.382 - 0.661i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.72 + 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.66T + 83T^{2} \)
89 \( 1 + (-8.01 - 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.87 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.81705973317230451163676817634, −10.24287165753236803379052103506, −8.674037444745396757160507675114, −7.915037686428110807881867915033, −7.41164871155574916551059249445, −6.20325959415169850751128514612, −4.78987977491974463965324830109, −3.53184020583659957754294386384, −2.45494575670366612903584550670, −1.54501951332436919256316370814, 2.09982137798414405335413691955, 3.67358604037290854915818680371, 4.77901647395672804896443977754, 5.19059935917495014728301096912, 6.77287679993406250962737818204, 7.36242149676066909149215879190, 8.694058622198976913523165167547, 9.447454945820396185299020536163, 10.14058054645904977311896935158, 11.22025264930318871543493444973

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