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Properties

Label 2-950-95.9-c1-0-1
Degree $2$
Conductor $950$
Sign $0.395 - 0.918i$
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 + (−2.20 − 2.62i)3-s + (0.939 − 0.342i)4-s + (−2.62 − 2.20i)6-s + (−1.61 − 0.933i)7-s + (0.866 − 0.5i)8-s + (−1.52 + 8.62i)9-s + (1.80 + 3.12i)11-s + (−2.96 − 1.71i)12-s + (−1.82 + 2.17i)13-s + (−1.75 − 0.638i)14-s + (0.766 − 0.642i)16-s + (−5.89 + 1.03i)17-s + 8.75i·18-s + (−4.32 − 0.577i)19-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (−1.27 − 1.51i)3-s + (0.469 − 0.171i)4-s + (−1.07 − 0.899i)6-s + (−0.611 − 0.352i)7-s + (0.306 − 0.176i)8-s + (−0.506 + 2.87i)9-s + (0.543 + 0.941i)11-s + (−0.857 − 0.494i)12-s + (−0.505 + 0.602i)13-s + (−0.468 − 0.170i)14-s + (0.191 − 0.160i)16-s + (−1.43 + 0.252i)17-s + 2.06i·18-s + (−0.991 − 0.132i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.395 - 0.918i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.392979 + 0.258763i\)
\(L(\frac12)\) \(\approx\) \(0.392979 + 0.258763i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.32 + 0.577i)T \)
good3 \( 1 + (2.20 + 2.62i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (1.61 + 0.933i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.80 - 3.12i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.82 - 2.17i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (5.89 - 1.03i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-1.74 - 4.78i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.204 - 1.16i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.59 + 4.50i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.35iT - 37T^{2} \)
41 \( 1 + (2.85 - 2.39i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.229 - 0.631i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (6.22 + 1.09i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-0.876 - 2.40i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.827 - 4.69i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (7.73 - 2.81i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-7.84 - 1.38i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.81 - 0.659i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-3.22 - 3.83i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-0.460 + 0.386i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (1.64 + 0.951i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.755 + 0.633i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.184 + 0.0325i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.56681094228605705140478640867, −9.550494589552782921810653755813, −8.188345822065959617789587042034, −7.06300483455058404356962887814, −6.78203369410080366943842059329, −6.13623649611464013202201873422, −5.00008708682457985730542499289, −4.24552516808174634217084986439, −2.42254436655910279597855802398, −1.54734734158972317548886219218, 0.19729788047421246845329413771, 2.87292495132727356336885122001, 3.83540701319034018147850744807, 4.66082583715117609693154420364, 5.37972076537581398777306592346, 6.44426709198778185190477014040, 6.53671319940018628247775488900, 8.508400332963484603117265388338, 9.207229847028997445174345642224, 10.10792424737778550329678500757

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