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Properties

Label 2-950-95.48-c1-0-25
Degree $2$
Conductor $950$
Sign $-0.989 - 0.147i$
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  + (0.573 − 0.819i)2-s + (0.133 − 1.52i)3-s + (−0.342 − 0.939i)4-s + (−1.17 − 0.984i)6-s + (−0.227 − 0.848i)7-s + (−0.965 − 0.258i)8-s + (0.642 + 0.113i)9-s + (−1.39 − 2.40i)11-s + (−1.47 + 0.396i)12-s + (−5.85 + 0.512i)13-s + (−0.825 − 0.300i)14-s + (−0.766 + 0.642i)16-s + (−0.437 − 0.306i)17-s + (0.461 − 0.461i)18-s + (3.66 − 2.35i)19-s + ⋯
L(s)  = 1  + (0.405 − 0.579i)2-s + (0.0770 − 0.881i)3-s + (−0.171 − 0.469i)4-s + (−0.479 − 0.402i)6-s + (−0.0859 − 0.320i)7-s + (−0.341 − 0.0915i)8-s + (0.214 + 0.0377i)9-s + (−0.419 − 0.726i)11-s + (−0.427 + 0.114i)12-s + (−1.62 + 0.142i)13-s + (−0.220 − 0.0802i)14-s + (−0.191 + 0.160i)16-s + (−0.106 − 0.0742i)17-s + (0.108 − 0.108i)18-s + (0.841 − 0.539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.989 - 0.147i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.989 - 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102084 + 1.37683i\)
\(L(\frac12)\) \(\approx\) \(0.102084 + 1.37683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.573 + 0.819i)T \)
5 \( 1 \)
19 \( 1 + (-3.66 + 2.35i)T \)
good3 \( 1 + (-0.133 + 1.52i)T + (-2.95 - 0.520i)T^{2} \)
7 \( 1 + (0.227 + 0.848i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.39 + 2.40i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.85 - 0.512i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (0.437 + 0.306i)T + (5.81 + 15.9i)T^{2} \)
23 \( 1 + (5.86 + 2.73i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.259 - 1.47i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.95 + 2.28i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.22 - 4.22i)T + 37iT^{2} \)
41 \( 1 + (-2.19 - 2.61i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.10 + 6.64i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (-1.04 - 1.49i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (-1.40 + 3.01i)T + (-34.0 - 40.6i)T^{2} \)
59 \( 1 + (-1.12 - 6.39i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (0.643 - 0.234i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.786 + 0.550i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (1.03 - 2.83i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (9.60 + 0.840i)T + (71.8 + 12.6i)T^{2} \)
79 \( 1 + (1.70 - 1.42i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-12.7 + 3.41i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (10.4 + 8.78i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-3.87 + 5.53i)T + (-33.1 - 91.1i)T^{2} \)
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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

−9.879764265974581769598352601265, −8.800164278046086180562736801877, −7.68900701850383559087703487077, −7.17566195112221025432058037286, −6.14714297832563281015995944467, −5.12925208083235735532278253675, −4.19363406592757181735569401955, −2.89016357768162554588386058809, −2.00998133390134536115662557048, −0.53165818888211182601071581401, 2.23023975760734653740955403073, 3.48028390932417994252907491051, 4.45806228583820642367575834285, 5.13320410280293329194542719818, 5.98308872161829804967662086108, 7.34818682273832514486573487848, 7.62468943945739209574325482438, 8.933797460587991137555308364959, 9.833859035859042645147794981065, 10.02670430141437717854974466409

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