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Properties

Label 2-950-95.9-c1-0-26
Degree $2$
Conductor $950$
Sign $-0.821 + 0.570i$
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 + (−0.984 − 1.17i)3-s + (0.939 − 0.342i)4-s + (−1.17 − 0.984i)6-s + (−1.62 − 0.939i)7-s + (0.866 − 0.5i)8-s + (0.113 − 0.642i)9-s + (0.0812 + 0.140i)11-s + (−1.32 − 0.766i)12-s + (3.03 − 3.61i)13-s + (−1.76 − 0.642i)14-s + (0.766 − 0.642i)16-s + (−4.28 + 0.754i)17-s − 0.652i·18-s + (−2.77 + 3.35i)19-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (−0.568 − 0.677i)3-s + (0.469 − 0.171i)4-s + (−0.479 − 0.402i)6-s + (−0.615 − 0.355i)7-s + (0.306 − 0.176i)8-s + (0.0377 − 0.214i)9-s + (0.0244 + 0.0424i)11-s + (−0.383 − 0.221i)12-s + (0.840 − 1.00i)13-s + (−0.471 − 0.171i)14-s + (0.191 − 0.160i)16-s + (−1.03 + 0.183i)17-s − 0.153i·18-s + (−0.637 + 0.770i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.821 + 0.570i$
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: \((2,\ 950,\ (\ :1/2),\ -0.821 + 0.570i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421939 - 1.34796i\)
\(L(\frac12)\) \(\approx\) \(0.421939 - 1.34796i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.77 - 3.35i)T \)
good3 \( 1 + (0.984 + 1.17i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (1.62 + 0.939i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0812 - 0.140i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.03 + 3.61i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (4.28 - 0.754i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (2.08 + 5.73i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.414 - 2.35i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.81 + 3.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.36iT - 37T^{2} \)
41 \( 1 + (-1.81 + 1.52i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.223 + 0.613i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (6.77 + 1.19i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (2.03 + 5.58i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.708 - 4.01i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.21 - 0.805i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.61 + 0.284i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-2.39 - 0.872i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-10.0 - 12.0i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.885 - 0.511i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.57 - 2.15i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-17.6 + 3.10i)T + (91.1 - 33.1i)T^{2} \)
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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

−9.986109448582296987175037534953, −8.786315894771241362828903545521, −7.85310162517819110934752362585, −6.74904452247989259020094979565, −6.31685075892942449119153726934, −5.55528147519313003515473991821, −4.23357076335463976121774603619, −3.43537975047543954493415257882, −2.04492629051289306725751449667, −0.53805301743269756364513096264, 1.99864253252890134436098936260, 3.32130969732453030128072520836, 4.37154967303578871908931118493, 4.95885159246537253563210238107, 6.16952095258834265629333530756, 6.51598316882518027960241191374, 7.75071905725878725106005475968, 8.871024050745097588976096687138, 9.583274941790227230106182515611, 10.55428527500667727613919523172

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