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Properties

Label 2-950-19.9-c1-0-4
Degree $2$
Conductor $950$
Sign $-0.997 - 0.0698i$
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  + (0.173 + 0.984i)2-s + (−1.52 + 1.28i)3-s + (−0.939 + 0.342i)4-s + (−1.52 − 1.28i)6-s + (1.97 − 3.41i)7-s + (−0.5 − 0.866i)8-s + (0.167 − 0.951i)9-s + (2.04 + 3.54i)11-s + (0.995 − 1.72i)12-s + (−1.91 − 1.60i)13-s + (3.70 + 1.34i)14-s + (0.766 − 0.642i)16-s + (0.884 + 5.01i)17-s + 0.966·18-s + (−1.12 + 4.21i)19-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (−0.880 + 0.739i)3-s + (−0.469 + 0.171i)4-s + (−0.622 − 0.522i)6-s + (0.745 − 1.29i)7-s + (−0.176 − 0.306i)8-s + (0.0559 − 0.317i)9-s + (0.616 + 1.06i)11-s + (0.287 − 0.497i)12-s + (−0.531 − 0.446i)13-s + (0.990 + 0.360i)14-s + (0.191 − 0.160i)16-s + (0.214 + 1.21i)17-s + 0.227·18-s + (−0.257 + 0.966i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.997 - 0.0698i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.997 - 0.0698i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0303934 + 0.868668i\)
\(L(\frac12)\) \(\approx\) \(0.0303934 + 0.868668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 \)
19 \( 1 + (1.12 - 4.21i)T \)
good3 \( 1 + (1.52 - 1.28i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-1.97 + 3.41i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.04 - 3.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.91 + 1.60i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.884 - 5.01i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (3.39 - 1.23i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.962 - 5.45i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.07 - 3.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.80T + 37T^{2} \)
41 \( 1 + (-8.57 + 7.19i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.70 - 1.34i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.680 - 3.85i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-5.31 + 1.93i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.36 - 7.74i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (10.6 - 3.87i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.172 + 0.977i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.51 + 0.916i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (3.18 - 2.67i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (2.85 - 2.39i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (2.87 - 4.97i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (9.98 + 8.37i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.982 - 5.57i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.42770108938911797710443730713, −9.892756435898695460103592352577, −8.692718013381012216858282556372, −7.65767257301366478014148598853, −7.16803103105010200165460084917, −6.00166527180349690610243898040, −5.23313818434339044914000401162, −4.30466778982752410119364597465, −3.87159155880227964946989635570, −1.56624774848499323704675036168, 0.45075727584976732761458814553, 1.86469170387091651743753192363, 2.87308542900039266396038781052, 4.37131157987595267053384114104, 5.39677897721731972037837678565, 5.96349656738331290309387474950, 6.93086264288635381491324795803, 8.037916271224378989180868180138, 9.005095640277404900579869240885, 9.495450337075570461743184032168

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