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Properties

Label 2-475-95.64-c1-0-10
Degree $2$
Conductor $475$
Sign $0.224 - 0.974i$
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  + (−1.03 + 0.595i)2-s + (−2.63 + 1.52i)3-s + (−0.290 + 0.503i)4-s + (1.81 − 3.14i)6-s − 0.609i·7-s − 3.07i·8-s + (3.14 − 5.44i)9-s + 4.48·11-s − 1.77i·12-s + (3.84 + 2.21i)13-s + (0.362 + 0.628i)14-s + (1.24 + 2.16i)16-s + (2.51 − 1.45i)17-s + 7.48i·18-s + (−3.60 − 2.44i)19-s + ⋯
L(s)  = 1  + (−0.729 + 0.421i)2-s + (−1.52 + 0.879i)3-s + (−0.145 + 0.251i)4-s + (0.740 − 1.28i)6-s − 0.230i·7-s − 1.08i·8-s + (1.04 − 1.81i)9-s + 1.35·11-s − 0.511i·12-s + (1.06 + 0.615i)13-s + (0.0969 + 0.167i)14-s + (0.312 + 0.540i)16-s + (0.609 − 0.352i)17-s + 1.76i·18-s + (−0.827 − 0.562i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.224 - 0.974i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.224 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.440275 + 0.350481i\)
\(L(\frac12)\) \(\approx\) \(0.440275 + 0.350481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 + 2.44i)T \)
good2 \( 1 + (1.03 - 0.595i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (2.63 - 1.52i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 0.609iT - 7T^{2} \)
11 \( 1 - 4.48T + 11T^{2} \)
13 \( 1 + (-3.84 - 2.21i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.51 + 1.45i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.46 + 1.42i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.558 + 0.966i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.22T + 31T^{2} \)
37 \( 1 + 3.77iT - 37T^{2} \)
41 \( 1 + (-4.15 - 7.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.65 + 4.99i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.09 - 2.94i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.31 - 4.22i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.11 - 8.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.49 + 4.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.34 + 4.23i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.80 + 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.22 - 1.86i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.51 - 7.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.12iT - 83T^{2} \)
89 \( 1 + (-3.96 + 6.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.37 - 4.83i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.03716324954487320041739906176, −10.34008114163438070390649812356, −9.272994349668942503203751891456, −8.908547606037295276387310731011, −7.37934700706401592363385555604, −6.50291122742265907313357472181, −5.82261112973514762878272440126, −4.27262156910737323016504146322, −3.90804658632750540814569214563, −0.873098489761392680247066256778, 0.878416261434917805901092019380, 1.81795457569367428111751856794, 4.07656092585295539926620894412, 5.69265782958563979745645963321, 5.87261021813492542971529870594, 7.05267120222335644000922226847, 8.188033844725970722625834400339, 9.071046958394386145893809487525, 10.26582492884395835113025342489, 10.82445533398250385570914986211

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