↑

Properties

Label 2-475-95.8-c1-0-3
Degree $2$
Conductor $475$
Sign $-0.691 + 0.722i$
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  + (−0.965 + 0.258i)2-s + (0.826 + 3.08i)3-s + (−0.866 + 0.5i)4-s + (−1.59 − 2.76i)6-s + (1.22 − 1.22i)7-s + (2.12 − 2.12i)8-s + (−6.22 + 3.59i)9-s − 4.19·11-s + (−2.25 − 2.25i)12-s + (−4.04 − 1.08i)13-s + (−0.866 + 1.49i)14-s + (−0.500 + 0.866i)16-s + (1.43 + 5.34i)17-s + (5.08 − 5.08i)18-s + (−4.33 + 0.5i)19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.477 + 1.78i)3-s + (−0.433 + 0.250i)4-s + (−0.651 − 1.12i)6-s + (0.462 − 0.462i)7-s + (0.749 − 0.749i)8-s + (−2.07 + 1.19i)9-s − 1.26·11-s + (−0.651 − 0.651i)12-s + (−1.12 − 0.300i)13-s + (−0.231 + 0.400i)14-s + (−0.125 + 0.216i)16-s + (0.347 + 1.29i)17-s + (1.19 − 1.19i)18-s + (−0.993 + 0.114i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.178815 - 0.418754i\)
\(L(\frac12)\) \(\approx\) \(0.178815 - 0.418754i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.33 - 0.5i)T \)
good2 \( 1 + (0.965 - 0.258i)T + (1.73 - i)T^{2} \)
3 \( 1 + (-0.826 - 3.08i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.22 + 1.22i)T - 7iT^{2} \)
11 \( 1 + 4.19T + 11T^{2} \)
13 \( 1 + (4.04 + 1.08i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.43 - 5.34i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (-0.448 + 1.67i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.76 + 4.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.06iT - 31T^{2} \)
37 \( 1 + (-1.68 - 1.68i)T + 37iT^{2} \)
41 \( 1 + (-3.28 - 1.89i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.322 + 0.0863i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-9.00 - 2.41i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.76 - 1.81i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.66 - 8.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.28 + 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.78 - 10.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.07 + 1.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (12.0 - 3.22i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.69 - 9.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.13 - 5.13i)T + 83iT^{2} \)
89 \( 1 + (-2.59 - 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.94 - 1.86i)T + (84.0 - 48.5i)T^{2} \)
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

−10.87048451086117351063968829522, −10.41260452190457828825885324123, −9.882380481028302731932582638681, −8.929952177503928629140098024386, −8.152898463003003283389712860333, −7.59850212557185723565530648988, −5.65304147076123057019300002224, −4.58511669850463640968074049840, −4.05873066673567553763557211226, −2.68600252322775618876235667393, 0.31582345473571152912202000696, 1.91146925530974919069288124650, 2.67565153115607896854094510511, 4.91257890994007051440342356435, 5.80225082938390962509719493813, 7.36006479045875998386906862429, 7.57566463076104301516669944183, 8.640601955400674387281153235802, 9.204565186902791866378624524485, 10.39444843280074743985603790425

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