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Properties

Label 2-475-95.88-c1-0-16
Degree $2$
Conductor $475$
Sign $0.941 + 0.336i$
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  + (0.258 − 0.965i)2-s + (2.11 + 0.567i)3-s + (0.866 + 0.5i)4-s + (1.09 − 1.89i)6-s + (−1.22 + 1.22i)7-s + (2.12 − 2.12i)8-s + (1.56 + 0.903i)9-s + 1.19·11-s + (1.55 + 1.55i)12-s + (−0.308 − 1.15i)13-s + (0.866 + 1.49i)14-s + (−0.500 − 0.866i)16-s + (−3.66 − 0.982i)17-s + (1.27 − 1.27i)18-s + (4.33 + 0.5i)19-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (1.22 + 0.327i)3-s + (0.433 + 0.250i)4-s + (0.447 − 0.775i)6-s + (−0.462 + 0.462i)7-s + (0.749 − 0.749i)8-s + (0.521 + 0.301i)9-s + 0.359·11-s + (0.447 + 0.447i)12-s + (−0.0856 − 0.319i)13-s + (0.231 + 0.400i)14-s + (−0.125 − 0.216i)16-s + (−0.889 − 0.238i)17-s + (0.301 − 0.301i)18-s + (0.993 + 0.114i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (468, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.941 + 0.336i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48491 - 0.431040i\)
\(L(\frac12)\) \(\approx\) \(2.48491 - 0.431040i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.33 - 0.5i)T \)
good2 \( 1 + (-0.258 + 0.965i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-2.11 - 0.567i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.22 - 1.22i)T - 7iT^{2} \)
11 \( 1 - 1.19T + 11T^{2} \)
13 \( 1 + (0.308 + 1.15i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.66 + 0.982i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (-1.67 + 0.448i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.89 - 3.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.26iT - 31T^{2} \)
37 \( 1 + (5.92 + 5.92i)T + 37iT^{2} \)
41 \( 1 + (4.78 - 2.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.32 - 8.68i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.41 + 9.00i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.81 + 6.76i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.66 + 8.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.78 + 3.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.4 - 2.78i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-14.0 + 8.12i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.810 - 3.02i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.29 + 14.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.46 - 1.46i)T + 83iT^{2} \)
89 \( 1 + (2.59 - 4.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.467 + 1.74i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.98748694560541801107787879766, −9.993789887719666250246721754183, −9.240513258816019662930761928291, −8.463263839011576927668077067219, −7.40517730518704525010232472477, −6.49476926165731832624657729562, −4.93025329570086053728266806747, −3.51922679189341349949923933509, −3.06967039072676474051798515561, −1.88653738189998416217042891995, 1.76451144458822681596853554167, 2.95514662714792079005861793335, 4.20775027114242589987010973226, 5.58584721667832899737922297293, 6.72962490837956185388190367139, 7.29940177963338318925871984988, 8.178551533838324289528067641323, 9.097628727292894927470087989534, 9.971889641505074675673103027690, 11.05620219614408632368953038612

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