Properties

Label 2-475-475.111-c1-0-45
Degree $2$
Conductor $475$
Sign $-0.986 - 0.166i$
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  + (0.0694 + 0.00485i)2-s + (−0.0877 − 2.51i)3-s + (−1.97 − 0.277i)4-s + (1.38 − 1.75i)5-s + (0.00611 − 0.175i)6-s + (−2.47 − 4.29i)7-s + (−0.272 − 0.0578i)8-s + (−3.32 + 0.232i)9-s + (0.104 − 0.115i)10-s + (0.0600 + 0.571i)11-s + (−0.524 + 4.99i)12-s + (2.98 + 4.43i)13-s + (−0.151 − 0.310i)14-s + (−4.53 − 3.32i)15-s + (3.81 + 1.09i)16-s + (−1.04 + 2.58i)17-s + ⋯
L(s)  = 1  + (0.0491 + 0.00343i)2-s + (−0.0506 − 1.45i)3-s + (−0.987 − 0.138i)4-s + (0.618 − 0.785i)5-s + (0.00249 − 0.0714i)6-s + (−0.937 − 1.62i)7-s + (−0.0962 − 0.0204i)8-s + (−1.10 + 0.0773i)9-s + (0.0330 − 0.0364i)10-s + (0.0181 + 0.172i)11-s + (−0.151 + 1.44i)12-s + (0.829 + 1.22i)13-s + (−0.0404 − 0.0829i)14-s + (−1.17 − 0.858i)15-s + (0.954 + 0.273i)16-s + (−0.253 + 0.626i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.986 - 0.166i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.986 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0783206 + 0.935321i\)
\(L(\frac12)\) \(\approx\) \(0.0783206 + 0.935321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.38 + 1.75i)T \)
19 \( 1 + (2.36 + 3.66i)T \)
good2 \( 1 + (-0.0694 - 0.00485i)T + (1.98 + 0.278i)T^{2} \)
3 \( 1 + (0.0877 + 2.51i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (2.47 + 4.29i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0600 - 0.571i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (-2.98 - 4.43i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (1.04 - 2.58i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (-5.62 - 5.43i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (1.69 + 4.18i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (-3.25 + 3.61i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-1.00 - 0.730i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (11.1 + 3.19i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (-0.909 + 5.15i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.33 - 3.29i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (5.11 + 0.718i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (0.457 + 1.83i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (2.11 + 2.04i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (-6.46 - 4.03i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (1.08 + 0.574i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (-4.05 + 6.00i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (0.322 + 9.22i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (-4.26 + 4.73i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-2.71 + 0.777i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (1.31 - 0.824i)T + (42.5 - 87.1i)T^{2} \)
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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

−10.42530533345456100245656225786, −9.475513510205236745943071755162, −8.796932387902633449460009274043, −7.72195280732782410729911548345, −6.72647824179634615945062391096, −6.14969818030269710204710156231, −4.69410315730796626498051936987, −3.73686404172305141834023613749, −1.66119896063408372464686395233, −0.60827978144604160385991223115, 2.93502875962188421905061634518, 3.43809350183326536312813045473, 4.95926667530131204054497357447, 5.61287908442756365642193680159, 6.49040805699168703258684509958, 8.442266464996833243423958779367, 8.936567150885227629275382053005, 9.764319530017553940481236611774, 10.30347994942679901380830904760, 11.16572346214404895262016623461

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