↑

Properties

Label 2-425-17.10-c2-0-53
Degree $2$
Conductor $425$
Sign $0.806 - 0.591i$
Analytic cond. $11.5804$
Root an. cond. $3.40300$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.42 − 3.45i)2-s + (−2.42 − 1.61i)3-s + (−7.03 − 7.03i)4-s + (−9.05 + 6.04i)6-s + (−8.53 − 1.69i)7-s + (−20.5 + 8.50i)8-s + (−0.194 − 0.468i)9-s + (1.94 + 2.90i)11-s + (5.65 + 28.4i)12-s + (11.5 − 11.5i)13-s + (−18.0 + 27.0i)14-s + 43.2i·16-s + (10.9 − 13.0i)17-s − 1.89·18-s + (−6.12 + 14.7i)19-s + ⋯
L(s)  = 1  + (0.714 − 1.72i)2-s + (−0.807 − 0.539i)3-s + (−1.75 − 1.75i)4-s + (−1.50 + 1.00i)6-s + (−1.21 − 0.242i)7-s + (−2.56 + 1.06i)8-s + (−0.0215 − 0.0520i)9-s + (0.176 + 0.264i)11-s + (0.471 + 2.37i)12-s + (0.888 − 0.888i)13-s + (−1.29 + 1.93i)14-s + 2.70i·16-s + (0.642 − 0.766i)17-s − 0.105·18-s + (−0.322 + 0.778i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.806 - 0.591i$
Analytic conductor: \(11.5804\)
Root analytic conductor: \(3.40300\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1),\ 0.806 - 0.591i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.549553 + 0.179792i\)
\(L(\frac12)\) \(\approx\) \(0.549553 + 0.179792i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (-10.9 + 13.0i)T \)
good2 \( 1 + (-1.42 + 3.45i)T + (-2.82 - 2.82i)T^{2} \)
3 \( 1 + (2.42 + 1.61i)T + (3.44 + 8.31i)T^{2} \)
7 \( 1 + (8.53 + 1.69i)T + (45.2 + 18.7i)T^{2} \)
11 \( 1 + (-1.94 - 2.90i)T + (-46.3 + 111. i)T^{2} \)
13 \( 1 + (-11.5 + 11.5i)T - 169iT^{2} \)
19 \( 1 + (6.12 - 14.7i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-15.7 + 10.5i)T + (202. - 488. i)T^{2} \)
29 \( 1 + (0.161 + 0.809i)T + (-776. + 321. i)T^{2} \)
31 \( 1 + (27.1 - 40.6i)T + (-367. - 887. i)T^{2} \)
37 \( 1 + (12.7 + 8.48i)T + (523. + 1.26e3i)T^{2} \)
41 \( 1 + (-7.15 - 1.42i)T + (1.55e3 + 643. i)T^{2} \)
43 \( 1 + (29.6 + 71.5i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + (6.97 - 6.97i)T - 2.20e3iT^{2} \)
53 \( 1 + (20.2 - 48.9i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (18.5 - 7.69i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-6.09 + 30.6i)T + (-3.43e3 - 1.42e3i)T^{2} \)
67 \( 1 + 32.7iT - 4.48e3T^{2} \)
71 \( 1 + (-47.1 - 31.4i)T + (1.92e3 + 4.65e3i)T^{2} \)
73 \( 1 + (132. - 26.3i)T + (4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (-33.2 - 49.8i)T + (-2.38e3 + 5.76e3i)T^{2} \)
83 \( 1 + (-90.0 - 37.3i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (65.9 + 65.9i)T + 7.92e3iT^{2} \)
97 \( 1 + (-3.04 - 15.3i)T + (-8.69e3 + 3.60e3i)T^{2} \)
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

−10.51222862452803652195684023975, −9.703624820533326643915396091488, −8.771274144075843280824539495712, −6.99099572547306393479588823747, −5.96997210974618236939448669350, −5.21359814563392497618504725962, −3.71606532563121785324498933747, −3.05726474539218559873645189714, −1.33770340225415508974612690523, −0.23484523400907617008616750919, 3.38398989785528769161288763275, 4.31271882697915890296370604582, 5.38009802249503813069851188021, 6.17114742392160487514550572134, 6.64351301936531768847423337659, 7.87582063216970033364566830230, 8.915411151151446968259447855848, 9.649117276814609025285649931622, 10.98461568243464298772191367737, 11.92434226254468286630278308365

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