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Properties

Label 2-950-95.74-c1-0-24
Degree $2$
Conductor $950$
Sign $0.855 + 0.517i$
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  + (0.984 + 0.173i)2-s + (1.07 − 1.28i)3-s + (0.939 + 0.342i)4-s + (1.28 − 1.07i)6-s + (−0.411 + 0.237i)7-s + (0.866 + 0.5i)8-s + (0.0320 + 0.181i)9-s + (2.79 − 4.84i)11-s + (1.45 − 0.838i)12-s + (1.58 + 1.88i)13-s + (−0.446 + 0.162i)14-s + (0.766 + 0.642i)16-s + (2.82 + 0.498i)17-s + 0.184i·18-s + (−3.17 + 2.98i)19-s + ⋯
L(s)  = 1  + (0.696 + 0.122i)2-s + (0.622 − 0.742i)3-s + (0.469 + 0.171i)4-s + (0.524 − 0.440i)6-s + (−0.155 + 0.0898i)7-s + (0.306 + 0.176i)8-s + (0.0106 + 0.0606i)9-s + (0.843 − 1.46i)11-s + (0.419 − 0.242i)12-s + (0.438 + 0.523i)13-s + (−0.119 + 0.0434i)14-s + (0.191 + 0.160i)16-s + (0.686 + 0.121i)17-s + 0.0435i·18-s + (−0.729 + 0.683i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.855 + 0.517i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.855 + 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.00870 - 0.839704i\)
\(L(\frac12)\) \(\approx\) \(3.00870 - 0.839704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 - 0.173i)T \)
5 \( 1 \)
19 \( 1 + (3.17 - 2.98i)T \)
good3 \( 1 + (-1.07 + 1.28i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.411 - 0.237i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.79 + 4.84i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.58 - 1.88i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.82 - 0.498i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-1.50 + 4.12i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.51 + 8.59i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.88 + 4.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.02iT - 37T^{2} \)
41 \( 1 + (-5.18 - 4.34i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.157 + 0.433i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (2.59 - 0.456i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (2.70 - 7.43i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.42 + 8.06i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (2.98 + 1.08i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.148 + 0.0262i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.48 - 1.63i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (5.91 - 7.04i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (9.48 + 7.96i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (11.4 - 6.59i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.23 + 1.03i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (4.95 + 0.873i)T + (91.1 + 33.1i)T^{2} \)
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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

−9.961567624220193393083488556416, −8.830981572285257872004919188046, −8.219100116772308314578261617513, −7.47115373766254007544966115618, −6.22604377932635190331537435928, −6.06599003293546113232684546505, −4.50329038581990343371327415590, −3.56887563416262045035272874165, −2.58341700347907423048935353340, −1.34409834835157540255147584946, 1.61693353626970437269319067048, 3.05672210477443646999409371623, 3.79269551947147906469223811011, 4.61389491237706127173073833832, 5.55864444316914731347363193540, 6.78623236701459847972680152334, 7.35185051282475349305793278823, 8.719047025949038433481391940903, 9.324408764918118485663159089774, 10.11242180250174194632951609894

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