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Properties

Label 2-950-19.9-c1-0-9
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

\(L(1)\) \(\approx\) \(1.71332 - 0.0599466i\)
\(L(\frac12)\) \(\approx\) \(1.71332 - 0.0599466i\)
\(L(\frac{3}{2})\) 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

−9.778263383553956882418215940028, −8.995422672793299318362122519560, −8.811557268666472706460665530000, −7.55075781145683806533178453787, −6.83913811906153297097158332769, −5.73763826447586835909499272795, −4.53053943278150998333783566477, −3.25526036554611081094280271117, −2.46745472952267365224256462582, −1.57669863578990644664142557540, 0.805684059894585978261269044777, 3.03926795824862761917785714078, 3.83365333572938995832755788795, 4.44089983499391253475589724337, 6.06616442899210051734010998246, 6.50432851786445856040763168349, 7.74058886458310527429159443067, 8.347605240367667350232536505510, 9.303380269481228107890475506584, 9.675953003908501798779659906699

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