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Properties

Label 2-950-19.17-c1-0-16
Degree $2$
Conductor $950$
Sign $0.114 + 0.993i$
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 + (−0.654 − 0.549i)3-s + (−0.939 − 0.342i)4-s + (−0.654 + 0.549i)6-s + (1.39 + 2.40i)7-s + (−0.5 + 0.866i)8-s + (−0.394 − 2.23i)9-s + (−1.58 + 2.74i)11-s + (0.427 + 0.739i)12-s + (3.68 − 3.08i)13-s + (2.61 − 0.951i)14-s + (0.766 + 0.642i)16-s + (−0.235 + 1.33i)17-s − 2.27·18-s + (4.15 − 1.31i)19-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.377 − 0.317i)3-s + (−0.469 − 0.171i)4-s + (−0.267 + 0.224i)6-s + (0.525 + 0.910i)7-s + (−0.176 + 0.306i)8-s + (−0.131 − 0.745i)9-s + (−0.477 + 0.826i)11-s + (0.123 + 0.213i)12-s + (1.02 − 0.856i)13-s + (0.698 − 0.254i)14-s + (0.191 + 0.160i)16-s + (−0.0571 + 0.324i)17-s − 0.535·18-s + (0.953 − 0.300i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.114 + 0.993i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10824 - 0.987386i\)
\(L(\frac12)\) \(\approx\) \(1.10824 - 0.987386i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.15 + 1.31i)T \)
good3 \( 1 + (0.654 + 0.549i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-1.39 - 2.40i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.58 - 2.74i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.68 + 3.08i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.235 - 1.33i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-5.66 - 2.06i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.81 + 10.2i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (3.80 + 6.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + (-2.86 - 2.40i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-1.44 + 0.525i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.516 - 2.92i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (4.04 + 1.47i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.714 + 4.04i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-3.98 - 1.44i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.992 + 5.63i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (0.844 - 0.307i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (8.01 + 6.72i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (1.71 + 1.43i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-4.16 - 7.21i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.26 - 6.09i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.456 + 2.58i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.729404515334922326320036798004, −9.281830806254042287140967390864, −8.204818481973120866863407603917, −7.45813297994868176031775496818, −6.04925028653910620948534580005, −5.60546833252030729181765996781, −4.49867251257710972780718066295, −3.30721246271063093110439419144, −2.23985109256568146313098871073, −0.885553230530383293243628603832, 1.18743959757748917731812677326, 3.13843580417850830909694998407, 4.24450650560412137427353436556, 5.08295864763266496143549909176, 5.78656696254429531256989602294, 6.94219261388878749051215806079, 7.57631409867589814983950054666, 8.515889812010511767878134369474, 9.187967089179842191136257640445, 10.45108097871979201834975984222

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