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Properties

Label 2-475-19.11-c1-0-15
Degree $2$
Conductor $475$
Sign $0.655 + 0.755i$
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  + (0.740 − 1.28i)2-s + (0.0908 − 0.157i)3-s + (−0.0969 − 0.167i)4-s + (−0.134 − 0.232i)6-s − 1.30·7-s + 2.67·8-s + (1.48 + 2.56i)9-s + 4.98·11-s − 0.0352·12-s + (0.203 + 0.351i)13-s + (−0.965 + 1.67i)14-s + (2.17 − 3.76i)16-s + (1.37 − 2.38i)17-s + 4.39·18-s + (−4.35 − 0.0955i)19-s + ⋯
L(s)  = 1  + (0.523 − 0.907i)2-s + (0.0524 − 0.0908i)3-s + (−0.0484 − 0.0839i)4-s + (−0.0549 − 0.0951i)6-s − 0.492·7-s + 0.945·8-s + (0.494 + 0.856i)9-s + 1.50·11-s − 0.0101·12-s + (0.0563 + 0.0975i)13-s + (−0.258 + 0.447i)14-s + (0.543 − 0.941i)16-s + (0.333 − 0.577i)17-s + 1.03·18-s + (−0.999 − 0.0219i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.655 + 0.755i$
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: \((2,\ 475,\ (\ :1/2),\ 0.655 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94452 - 0.887400i\)
\(L(\frac12)\) \(\approx\) \(1.94452 - 0.887400i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.35 + 0.0955i)T \)
good2 \( 1 + (-0.740 + 1.28i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.0908 + 0.157i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 1.30T + 7T^{2} \)
11 \( 1 - 4.98T + 11T^{2} \)
13 \( 1 + (-0.203 - 0.351i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.37 + 2.38i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.47 + 6.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.00 - 3.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.57T + 31T^{2} \)
37 \( 1 - 3.71T + 37T^{2} \)
41 \( 1 + (-0.607 + 1.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.56 - 2.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.25 - 5.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.16 + 5.47i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.61 - 9.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.467 + 0.808i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.64 + 4.58i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.817 + 1.41i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.84 - 6.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.27 + 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + (7.10 + 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.14 - 15.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.96580965636596278950322980409, −10.27180344400938729626010756896, −9.304593322032336632229757201818, −8.195022561223160123966831125031, −7.13873966014677926655071741802, −6.27311682514988582592674196787, −4.69297010501616871105268148245, −4.00099082697031471097229663204, −2.77101835865932690081705794350, −1.57318216110760876594210483812, 1.52734987891438424828430035882, 3.66835510561360995180920661237, 4.31865729961090625826213148827, 5.82557199429568837331335016103, 6.39624724836649779889770172940, 7.13053286013700905420601326187, 8.266371119638830772881463038706, 9.404404931488522472359316983062, 10.03936143948877775455808582500, 11.18057284716157059918194555135

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