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Properties

Label 2-950-95.67-c1-0-16
Degree $2$
Conductor $950$
Sign $0.970 - 0.239i$
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  + (−0.819 − 0.573i)2-s + (1.52 + 0.133i)3-s + (0.342 + 0.939i)4-s + (−1.17 − 0.984i)6-s + (3.99 − 1.06i)7-s + (0.258 − 0.965i)8-s + (−0.642 − 0.113i)9-s + (1.83 + 3.17i)11-s + (0.396 + 1.47i)12-s + (0.542 + 6.20i)13-s + (−3.88 − 1.41i)14-s + (−0.766 + 0.642i)16-s + (−1.30 + 1.86i)17-s + (0.461 + 0.461i)18-s + (4.31 + 0.639i)19-s + ⋯
L(s)  = 1  + (−0.579 − 0.405i)2-s + (0.881 + 0.0770i)3-s + (0.171 + 0.469i)4-s + (−0.479 − 0.402i)6-s + (1.50 − 0.404i)7-s + (0.0915 − 0.341i)8-s + (−0.214 − 0.0377i)9-s + (0.551 + 0.956i)11-s + (0.114 + 0.427i)12-s + (0.150 + 1.72i)13-s + (−1.03 − 0.377i)14-s + (−0.191 + 0.160i)16-s + (−0.316 + 0.451i)17-s + (0.108 + 0.108i)18-s + (0.989 + 0.146i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.970 - 0.239i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.970 - 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83490 + 0.222740i\)
\(L(\frac12)\) \(\approx\) \(1.83490 + 0.222740i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.819 + 0.573i)T \)
5 \( 1 \)
19 \( 1 + (-4.31 - 0.639i)T \)
good3 \( 1 + (-1.52 - 0.133i)T + (2.95 + 0.520i)T^{2} \)
7 \( 1 + (-3.99 + 1.06i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.83 - 3.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.542 - 6.20i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (1.30 - 1.86i)T + (-5.81 - 15.9i)T^{2} \)
23 \( 1 + (3.36 - 7.21i)T + (-14.7 - 17.6i)T^{2} \)
29 \( 1 + (-1.32 + 7.52i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (5.27 + 3.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.80 + 5.80i)T - 37iT^{2} \)
41 \( 1 + (3.86 + 4.60i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.65 + 1.23i)T + (27.6 - 32.9i)T^{2} \)
47 \( 1 + (4.79 - 3.35i)T + (16.0 - 44.1i)T^{2} \)
53 \( 1 + (-8.85 - 4.12i)T + (34.0 + 40.6i)T^{2} \)
59 \( 1 + (-1.07 - 6.09i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.57 + 0.938i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-6.39 - 9.13i)T + (-22.9 + 62.9i)T^{2} \)
71 \( 1 + (-2.18 + 6.01i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-0.304 + 3.47i)T + (-71.8 - 12.6i)T^{2} \)
79 \( 1 + (-6.41 + 5.37i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (2.46 + 9.20i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-0.289 - 0.242i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (11.7 + 8.25i)T + (33.1 + 91.1i)T^{2} \)
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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

−9.710160593118617587402089778253, −9.339918318864501233049147099364, −8.487811813418191944372643089583, −7.68628926633803488147861489350, −7.17136517076310331239597629767, −5.74579150266851609132444800772, −4.30137145844843774211955031909, −3.86949979172471536214149667714, −2.17810641717694956028036667274, −1.62090169969230097379183608394, 1.05453367768092065473444958919, 2.44533518721060077028522007496, 3.41187143326365390084730986625, 5.01699693942384769771380615906, 5.59371974000536977773810263934, 6.80391440000677218996013458193, 8.006288747305243743071678734730, 8.250460967091906109029552329472, 8.800857233567813095594960479863, 9.800388131412518677929403348685

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