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Properties

Label 2-950-19.4-c1-0-22
Degree $2$
Conductor $950$
Sign $0.724 + 0.689i$
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  + (0.939 + 0.342i)2-s + (−0.260 − 1.47i)3-s + (0.766 + 0.642i)4-s + (0.260 − 1.47i)6-s + (1.51 − 2.61i)7-s + (0.500 + 0.866i)8-s + (0.699 − 0.254i)9-s + (2.53 + 4.39i)11-s + (0.750 − 1.30i)12-s + (0.272 − 1.54i)13-s + (2.31 − 1.94i)14-s + (0.173 + 0.984i)16-s + (1.18 + 0.431i)17-s + 0.744·18-s + (−3.72 − 2.26i)19-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (−0.150 − 0.853i)3-s + (0.383 + 0.321i)4-s + (0.106 − 0.603i)6-s + (0.571 − 0.990i)7-s + (0.176 + 0.306i)8-s + (0.233 − 0.0848i)9-s + (0.764 + 1.32i)11-s + (0.216 − 0.375i)12-s + (0.0756 − 0.428i)13-s + (0.619 − 0.519i)14-s + (0.0434 + 0.246i)16-s + (0.287 + 0.104i)17-s + 0.175·18-s + (−0.854 − 0.519i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.724 + 0.689i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.724 + 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37739 - 0.949967i\)
\(L(\frac12)\) \(\approx\) \(2.37739 - 0.949967i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 \)
19 \( 1 + (3.72 + 2.26i)T \)
good3 \( 1 + (0.260 + 1.47i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-1.51 + 2.61i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.53 - 4.39i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.272 + 1.54i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.18 - 0.431i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-0.795 - 0.667i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-1.90 + 0.694i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-0.312 + 0.541i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.24T + 37T^{2} \)
41 \( 1 + (0.189 + 1.07i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-8.82 + 7.40i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-9.85 + 3.58i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (0.484 + 0.406i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-5.25 - 1.91i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (8.87 + 7.44i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (11.5 - 4.20i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (10.5 - 8.84i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-2.46 - 13.9i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.424 - 2.41i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.10 + 1.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.09 - 11.9i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-11.4 - 4.17i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.15815742410606327647815452112, −8.981315444027743781992400328859, −7.86179521570467941749850575411, −7.14855853332104045318531682025, −6.80449175719733540545561311709, −5.66496680662881763994078367662, −4.46932064047952546143078662072, −3.95058458705240446083585281365, −2.26658943035999945669025617644, −1.16637440145836889049660660467, 1.57699776633817924621202674677, 2.95305672391242443995075419148, 3.99753086342156018329331860823, 4.73579079517520707530143097561, 5.72456098441980835740707693806, 6.30376052427987318878723438970, 7.63521743747208858262240283621, 8.796636603896883446603686998755, 9.203443425039968953842868310308, 10.45934040625483641723874626704

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