Properties

Label 2-475-475.396-c1-0-9
Degree $2$
Conductor $475$
Sign $0.535 - 0.844i$
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  + (−0.978 + 0.611i)2-s + (−2.30 − 0.662i)3-s + (−0.293 + 0.601i)4-s + (2.15 + 0.608i)5-s + (2.66 − 0.764i)6-s + (0.629 + 1.09i)7-s + (−0.321 − 3.06i)8-s + (2.35 + 1.47i)9-s + (−2.47 + 0.719i)10-s + (0.499 + 0.555i)11-s + (1.07 − 1.19i)12-s + (0.209 − 5.98i)13-s + (−1.28 − 0.681i)14-s + (−4.56 − 2.83i)15-s + (1.36 + 1.74i)16-s + (−1.59 − 0.111i)17-s + ⋯
L(s)  = 1  + (−0.691 + 0.432i)2-s + (−1.33 − 0.382i)3-s + (−0.146 + 0.300i)4-s + (0.962 + 0.272i)5-s + (1.08 − 0.311i)6-s + (0.237 + 0.412i)7-s + (−0.113 − 1.08i)8-s + (0.784 + 0.490i)9-s + (−0.783 + 0.227i)10-s + (0.150 + 0.167i)11-s + (0.310 − 0.345i)12-s + (0.0579 − 1.66i)13-s + (−0.342 − 0.182i)14-s + (−1.17 − 0.731i)15-s + (0.340 + 0.436i)16-s + (−0.386 − 0.0270i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (396, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.535 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591254 + 0.325395i\)
\(L(\frac12)\) \(\approx\) \(0.591254 + 0.325395i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.15 - 0.608i)T \)
19 \( 1 + (1.77 - 3.98i)T \)
good2 \( 1 + (0.978 - 0.611i)T + (0.876 - 1.79i)T^{2} \)
3 \( 1 + (2.30 + 0.662i)T + (2.54 + 1.58i)T^{2} \)
7 \( 1 + (-0.629 - 1.09i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.499 - 0.555i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (-0.209 + 5.98i)T + (-12.9 - 0.906i)T^{2} \)
17 \( 1 + (1.59 + 0.111i)T + (16.8 + 2.36i)T^{2} \)
23 \( 1 + (-7.02 - 0.987i)T + (22.1 + 6.33i)T^{2} \)
29 \( 1 + (0.0907 - 0.00634i)T + (28.7 - 4.03i)T^{2} \)
31 \( 1 + (-5.43 - 2.42i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (0.862 + 2.65i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-4.92 - 6.30i)T + (-9.91 + 39.7i)T^{2} \)
43 \( 1 + (1.24 - 7.03i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.369 + 0.0258i)T + (46.5 - 6.54i)T^{2} \)
53 \( 1 + (4.48 - 9.20i)T + (-32.6 - 41.7i)T^{2} \)
59 \( 1 + (1.73 - 4.30i)T + (-42.4 - 40.9i)T^{2} \)
61 \( 1 + (-3.48 - 0.490i)T + (58.6 + 16.8i)T^{2} \)
67 \( 1 + (-1.28 + 5.14i)T + (-59.1 - 31.4i)T^{2} \)
71 \( 1 + (7.40 - 7.15i)T + (2.47 - 70.9i)T^{2} \)
73 \( 1 + (0.0977 + 2.79i)T + (-72.8 + 5.09i)T^{2} \)
79 \( 1 + (-15.2 - 4.37i)T + (66.9 + 41.8i)T^{2} \)
83 \( 1 + (-0.0853 - 0.0379i)T + (55.5 + 61.6i)T^{2} \)
89 \( 1 + (-6.15 + 7.87i)T + (-21.5 - 86.3i)T^{2} \)
97 \( 1 + (1.83 + 7.35i)T + (-85.6 + 45.5i)T^{2} \)
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

−10.93493965277187346748345785085, −10.32605183575080318305002815350, −9.373419206907832756091269900105, −8.395506800117782151049684025515, −7.41329381330140920415920929888, −6.45449310120034869946123091285, −5.81327001961978903520729864164, −4.84581171649317740088775900603, −3.00113842644595276842882094960, −1.06932534262609362367244280355, 0.814715820187990548939709328307, 2.16429509201687780418279823025, 4.51810219696228954145225542210, 5.06594152108923774036700170744, 6.16983024105780122024598295104, 6.87098176943730456009279699912, 8.682417108934254420769837285866, 9.254752661230362528768547287277, 10.08850214774102903775877610340, 10.94288178734817827546011998892

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