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Properties

Label 2-475-95.54-c1-0-7
Degree $2$
Conductor $475$
Sign $-0.963 - 0.269i$
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  + (−0.381 + 0.454i)2-s + (0.660 + 1.81i)3-s + (0.286 + 1.62i)4-s + (−1.07 − 0.392i)6-s + (−0.918 + 0.530i)7-s + (−1.87 − 1.08i)8-s + (−0.555 + 0.466i)9-s + (−0.0983 + 0.170i)11-s + (−2.75 + 1.58i)12-s + (−1.80 + 4.96i)13-s + (0.109 − 0.620i)14-s + (−1.88 + 0.686i)16-s + (0.453 − 0.540i)17-s − 0.430i·18-s + (4.24 + 0.983i)19-s + ⋯
L(s)  = 1  + (−0.269 + 0.321i)2-s + (0.381 + 1.04i)3-s + (0.143 + 0.811i)4-s + (−0.439 − 0.160i)6-s + (−0.347 + 0.200i)7-s + (−0.663 − 0.382i)8-s + (−0.185 + 0.155i)9-s + (−0.0296 + 0.0513i)11-s + (−0.794 + 0.458i)12-s + (−0.501 + 1.37i)13-s + (0.0292 − 0.165i)14-s + (−0.471 + 0.171i)16-s + (0.109 − 0.130i)17-s − 0.101i·18-s + (0.974 + 0.225i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.963 - 0.269i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.963 - 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162485 + 1.18550i\)
\(L(\frac12)\) \(\approx\) \(0.162485 + 1.18550i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.24 - 0.983i)T \)
good2 \( 1 + (0.381 - 0.454i)T + (-0.347 - 1.96i)T^{2} \)
3 \( 1 + (-0.660 - 1.81i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (0.918 - 0.530i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.0983 - 0.170i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.80 - 4.96i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.453 + 0.540i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (6.52 - 1.15i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.59 + 2.17i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (3.95 + 6.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.09iT - 37T^{2} \)
41 \( 1 + (-1.27 + 0.463i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-8.87 - 1.56i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (3.09 + 3.69i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-5.24 + 0.924i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-8.41 - 7.05i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.04 - 11.5i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.96 - 2.34i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.434 + 2.46i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (2.24 + 6.15i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (11.5 - 4.19i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (3.49 - 2.01i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-16.4 - 5.99i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (0.861 - 1.02i)T + (-16.8 - 95.5i)T^{2} \)
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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

−11.58453927216294038984558800294, −10.16704459155752093830976859835, −9.473550555470685134796680574938, −8.956528423787696795864996724119, −7.83309830810215195772406127701, −7.01006847342167569345365963564, −5.87604611867501808793169081486, −4.38152672825558952887221099522, −3.72721054320216282218550418815, −2.49053981526158423586457382805, 0.76099481858017047254847041004, 2.08513592022012409539539024587, 3.19965367755358471608857357165, 5.05389928558804872810333658682, 6.01257777370036097183183267182, 7.02409768395951021564805684336, 7.82110480346505886983604203625, 8.769975277158473261331658850389, 9.950441923509395824127688594463, 10.38013873490589784294679237819

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