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Properties

Label 2-950-475.111-c1-0-49
Degree $2$
Conductor $950$
Sign $0.564 - 0.825i$
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  + (−0.997 − 0.0697i)2-s + (−0.118 − 3.39i)3-s + (0.990 + 0.139i)4-s + (−2.19 + 0.440i)5-s + (−0.118 + 3.39i)6-s + (−1.36 − 2.36i)7-s + (−0.978 − 0.207i)8-s + (−8.51 + 0.595i)9-s + (2.21 − 0.286i)10-s + (0.292 + 2.77i)11-s + (0.355 − 3.37i)12-s + (−2.22 − 3.30i)13-s + (1.19 + 2.45i)14-s + (1.75 + 7.38i)15-s + (0.961 + 0.275i)16-s + (−0.119 + 0.295i)17-s + ⋯
L(s)  = 1  + (−0.705 − 0.0493i)2-s + (−0.0684 − 1.95i)3-s + (0.495 + 0.0695i)4-s + (−0.980 + 0.196i)5-s + (−0.0483 + 1.38i)6-s + (−0.516 − 0.895i)7-s + (−0.345 − 0.0735i)8-s + (−2.83 + 0.198i)9-s + (0.701 − 0.0904i)10-s + (0.0880 + 0.837i)11-s + (0.102 − 0.975i)12-s + (−0.618 − 0.916i)13-s + (0.320 + 0.656i)14-s + (0.452 + 1.90i)15-s + (0.240 + 0.0689i)16-s + (−0.0289 + 0.0715i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.564 - 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0402892 + 0.0212431i\)
\(L(\frac12)\) \(\approx\) \(0.0402892 + 0.0212431i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.997 + 0.0697i)T \)
5 \( 1 + (2.19 - 0.440i)T \)
19 \( 1 + (-0.392 + 4.34i)T \)
good3 \( 1 + (0.118 + 3.39i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (1.36 + 2.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.292 - 2.77i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (2.22 + 3.30i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (0.119 - 0.295i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (-3.03 - 2.93i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (1.66 + 4.12i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (6.27 - 6.96i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-8.72 - 6.33i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.68 + 0.770i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (-1.11 + 6.29i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.29 + 5.66i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (3.55 + 0.498i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (-2.41 - 9.68i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (2.51 + 2.43i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (8.43 + 5.27i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (-9.20 - 4.89i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (-0.129 + 0.191i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (-0.183 - 5.24i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (-3.46 + 3.85i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (2.39 - 0.686i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (3.68 - 2.30i)T + (42.5 - 87.1i)T^{2} \)
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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

−9.101361776621903131080091800735, −8.178040615278590330146929909778, −7.35677280383104472281936500481, −7.23365678296767718583656748191, −6.45769214211743393567341445818, −5.12583651765580289403754115087, −3.42392207862255878454457937511, −2.47797705009152625324089845902, −1.06270179581915028266095854357, −0.03163410137983162244992637535, 2.72110603449749493733154040566, 3.60134945390701767959566988581, 4.47467850920287549207607972131, 5.49982781488217261419171512415, 6.29459425670035101088947165381, 7.76107892871039270290267104255, 8.588683004257826060256372859380, 9.329684041774517607825905859207, 9.533404685179140804406716102768, 10.78685252783869259035845640069

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