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Properties

Label 2-950-25.6-c1-0-27
Degree $2$
Conductor $950$
Sign $-0.0953 + 0.995i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (−1.20 + 0.878i)3-s + (−0.809 + 0.587i)4-s + (0.176 − 2.22i)5-s + (1.20 + 0.878i)6-s + 2.76·7-s + (0.809 + 0.587i)8-s + (−0.236 + 0.728i)9-s + (−2.17 + 0.520i)10-s + (−0.936 − 2.88i)11-s + (0.461 − 1.42i)12-s + (0.408 − 1.25i)13-s + (−0.855 − 2.63i)14-s + (1.74 + 2.85i)15-s + (0.309 − 0.951i)16-s + (1.57 + 1.14i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.698 + 0.507i)3-s + (−0.404 + 0.293i)4-s + (0.0790 − 0.996i)5-s + (0.493 + 0.358i)6-s + 1.04·7-s + (0.286 + 0.207i)8-s + (−0.0789 + 0.242i)9-s + (−0.687 + 0.164i)10-s + (−0.282 − 0.869i)11-s + (0.133 − 0.410i)12-s + (0.113 − 0.349i)13-s + (−0.228 − 0.703i)14-s + (0.450 + 0.736i)15-s + (0.0772 − 0.237i)16-s + (0.382 + 0.278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0953 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0953 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0953 + 0.995i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.0953 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723293 - 0.795899i\)
\(L(\frac12)\) \(\approx\) \(0.723293 - 0.795899i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.176 + 2.22i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (1.20 - 0.878i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 2.76T + 7T^{2} \)
11 \( 1 + (0.936 + 2.88i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.408 + 1.25i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.57 - 1.14i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (-0.770 - 2.37i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.699 + 0.508i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-7.43 - 5.39i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.73 + 8.42i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.64 + 11.2i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.19T + 43T^{2} \)
47 \( 1 + (-6.42 + 4.66i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (8.85 - 6.43i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.78 + 11.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.993 + 3.05i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (6.53 + 4.75i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-12.2 + 8.88i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.95 - 6.02i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-11.9 + 8.66i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.52 + 4.01i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.98 + 15.3i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (12.2 - 8.87i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.02652855971671434071383910772, −9.010596056601766652264312234006, −8.268371036010526121321819191210, −7.72051674387403347642038345145, −5.95444833925879046465613876824, −5.22303845686876417563279874482, −4.66227891932122039130024831544, −3.52640497887601444204012794704, −1.97181668211570534556574326336, −0.68421061169579632776543023664, 1.28150881003973995384229684383, 2.73336111526834866555778384087, 4.33680199087674002777453868819, 5.20237940543322672781980646920, 6.26989591684020850207451057551, 6.75115568061122098781866572267, 7.64032007937505882678521948916, 8.248712628276747015418285705590, 9.555547134548343646726163664879, 10.15090894866811890120277284872

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