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Properties

Label 2-950-25.16-c1-0-3
Degree $2$
Conductor $950$
Sign $-0.516 - 0.856i$
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  + (−0.809 − 0.587i)2-s + (0.429 + 1.32i)3-s + (0.309 + 0.951i)4-s + (−1.05 − 1.97i)5-s + (0.429 − 1.32i)6-s − 3.34·7-s + (0.309 − 0.951i)8-s + (0.867 − 0.630i)9-s + (−0.305 + 2.21i)10-s + (1.04 + 0.755i)11-s + (−1.12 + 0.816i)12-s + (0.866 − 0.629i)13-s + (2.70 + 1.96i)14-s + (2.15 − 2.23i)15-s + (−0.809 + 0.587i)16-s + (−1.40 + 4.33i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.247 + 0.762i)3-s + (0.154 + 0.475i)4-s + (−0.471 − 0.881i)5-s + (0.175 − 0.539i)6-s − 1.26·7-s + (0.109 − 0.336i)8-s + (0.289 − 0.210i)9-s + (−0.0966 + 0.700i)10-s + (0.313 + 0.227i)11-s + (−0.324 + 0.235i)12-s + (0.240 − 0.174i)13-s + (0.722 + 0.525i)14-s + (0.555 − 0.578i)15-s + (−0.202 + 0.146i)16-s + (−0.341 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.516 - 0.856i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.516 - 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212889 + 0.377010i\)
\(L(\frac12)\) \(\approx\) \(0.212889 + 0.377010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (1.05 + 1.97i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.429 - 1.32i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 3.34T + 7T^{2} \)
11 \( 1 + (-1.04 - 0.755i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.866 + 0.629i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.40 - 4.33i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (5.68 + 4.13i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.10 - 6.46i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.10 - 6.48i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.66 - 4.11i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.72 + 3.43i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.32T + 43T^{2} \)
47 \( 1 + (-2.33 - 7.20i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.17 + 6.69i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (9.88 - 7.17i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (6.31 + 4.59i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (1.27 - 3.93i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.745 + 2.29i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.00 + 5.81i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.0685 + 0.211i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.99 - 15.3i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-9.31 - 6.76i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.594 - 1.82i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.28627907518010341715466347976, −9.422902338905239558625105930935, −8.873773235929304600179178080999, −8.201086596885806816427301382572, −6.98598655566708532048692823375, −6.15758696946792545555423117297, −4.73299047801448651409112649416, −3.86985863978379701445776220741, −3.22617955214851339202123382172, −1.47931756982074823655570220458, 0.23804150209895786400423680438, 2.07090303178317778343656897186, 3.13870072382360548374049449608, 4.28143202490060364800553897344, 5.98490242894998619205049902942, 6.50037077630547422422203714795, 7.35593976928983052204645652901, 7.75015939566649081985282165366, 8.904800933666087418457522434916, 9.736400152089312153007266548868

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