↑

Properties

Label 2-950-25.21-c1-0-24
Degree $2$
Conductor $950$
Sign $-0.321 + 0.946i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (−0.309 + 0.951i)2-s + (−2.00 − 1.45i)3-s + (−0.809 − 0.587i)4-s + (−2.21 − 0.295i)5-s + (2.00 − 1.45i)6-s + 2.78·7-s + (0.809 − 0.587i)8-s + (0.974 + 2.99i)9-s + (0.965 − 2.01i)10-s + (0.387 − 1.19i)11-s + (0.766 + 2.35i)12-s + (0.256 + 0.790i)13-s + (−0.859 + 2.64i)14-s + (4.01 + 3.82i)15-s + (0.309 + 0.951i)16-s + (3.13 − 2.27i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−1.15 − 0.841i)3-s + (−0.404 − 0.293i)4-s + (−0.991 − 0.132i)5-s + (0.819 − 0.595i)6-s + 1.05·7-s + (0.286 − 0.207i)8-s + (0.324 + 0.999i)9-s + (0.305 − 0.637i)10-s + (0.116 − 0.359i)11-s + (0.221 + 0.681i)12-s + (0.0712 + 0.219i)13-s + (−0.229 + 0.706i)14-s + (1.03 + 0.987i)15-s + (0.0772 + 0.237i)16-s + (0.760 − 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.321 + 0.946i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.321 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305091 - 0.425948i\)
\(L(\frac12)\) \(\approx\) \(0.305091 - 0.425948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (2.21 + 0.295i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (2.00 + 1.45i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 2.78T + 7T^{2} \)
11 \( 1 + (-0.387 + 1.19i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.256 - 0.790i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.13 + 2.27i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (1.45 - 4.47i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.32 + 1.68i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.29 + 3.12i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.24 + 3.84i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.52 + 7.76i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.51T + 43T^{2} \)
47 \( 1 + (3.42 + 2.48i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.232 - 0.169i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.22 + 9.92i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.37 + 7.30i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.70 + 6.32i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (7.37 + 5.36i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.33 - 4.10i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.6 + 7.70i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.6 - 8.47i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.45 - 4.48i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.870 + 0.632i)T + (29.9 + 92.2i)T^{2} \)
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

−9.747674496650401029701536934689, −8.604525322200203356054079016673, −7.83342377738117823541129531917, −7.33379867319140644720965211916, −6.47009022336338037324612309943, −5.45754807474355446864106963813, −4.89917406966161729505045399557, −3.65662321947294414642897778919, −1.56092908513024701842808140154, −0.36247692349029015648274755574, 1.29965902574623148972164414639, 3.11980523917504916398785452771, 4.28904525685136069864795329878, 4.72098324519755114669609394823, 5.69878517807962469065877114202, 6.93072306088137393088791054716, 8.055784592656322755884951723247, 8.538942348738146504591562137956, 9.956664591228166483229492959145, 10.36213104854774094113706595467

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