↑

Properties

Label 2-475-95.24-c1-0-13
Degree $2$
Conductor $475$
Sign $-0.234 - 0.972i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.691 + 1.89i)2-s + (1.87 + 0.330i)3-s + (−1.59 + 1.34i)4-s + (0.667 + 3.78i)6-s + (2.67 − 1.54i)7-s + (−0.152 − 0.0878i)8-s + (0.575 + 0.209i)9-s + (−0.481 + 0.834i)11-s + (−3.43 + 1.98i)12-s + (−2.91 + 0.513i)13-s + (4.78 + 4.01i)14-s + (−0.663 + 3.76i)16-s + (−0.0133 − 0.0366i)17-s + 1.23i·18-s + (4.31 − 0.596i)19-s + ⋯
L(s)  = 1  + (0.488 + 1.34i)2-s + (1.08 + 0.190i)3-s + (−0.799 + 0.670i)4-s + (0.272 + 1.54i)6-s + (1.01 − 0.584i)7-s + (−0.0538 − 0.0310i)8-s + (0.191 + 0.0697i)9-s + (−0.145 + 0.251i)11-s + (−0.991 + 0.572i)12-s + (−0.808 + 0.142i)13-s + (1.27 + 1.07i)14-s + (−0.165 + 0.940i)16-s + (−0.00323 − 0.00887i)17-s + 0.291i·18-s + (0.990 − 0.136i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.234 - 0.972i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.234 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64008 + 2.08174i\)
\(L(\frac12)\) \(\approx\) \(1.64008 + 2.08174i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.31 + 0.596i)T \)
good2 \( 1 + (-0.691 - 1.89i)T + (-1.53 + 1.28i)T^{2} \)
3 \( 1 + (-1.87 - 0.330i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (-2.67 + 1.54i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.481 - 0.834i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.91 - 0.513i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.0133 + 0.0366i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (2.49 + 2.97i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (8.76 + 3.19i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-4.68 - 8.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.11iT - 37T^{2} \)
41 \( 1 + (-2.09 + 11.8i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (3.18 - 3.79i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.74 + 7.53i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (3.89 + 4.64i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-2.60 + 0.948i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.68 - 6.44i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-3.19 + 8.78i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-9.64 - 8.09i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.883 + 0.155i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.54 - 8.75i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-8.31 + 4.80i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.51 - 8.61i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (2.66 + 7.31i)T + (-74.3 + 62.3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.24802326900642474015547981784, −10.17769651151580488350844371849, −9.132361367672617094887486759608, −8.205633576181306377746286736995, −7.63421347728629715529757207297, −6.91555104745914791985903148390, −5.52671858645167123591817231464, −4.69950973982327591135346333566, −3.71204553458530611114670413954, −2.11582813749039764011816196487, 1.66090962506565928416506103030, 2.56018383498196492744356500073, 3.44275406809688771584552173675, 4.67599956944928029689102071221, 5.64500978583971659219853698836, 7.56696812320953437202073689465, 8.003703664177413537655174374088, 9.263768119474888275474346797653, 9.800513392177520548314827440895, 11.10887940883350502134792838902

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