Properties

Label 2-950-19.7-c1-0-8
Degree $2$
Conductor $950$
Sign $0.955 + 0.295i$
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  + (−0.5 − 0.866i)2-s + (−1.28 − 2.21i)3-s + (−0.499 + 0.866i)4-s + (−1.28 + 2.21i)6-s − 0.438·7-s + 0.999·8-s + (−1.78 + 3.08i)9-s + 11-s + 2.56·12-s + (−1 + 1.73i)13-s + (0.219 + 0.379i)14-s + (−0.5 − 0.866i)16-s + (2.56 + 4.43i)17-s + 3.56·18-s + (−2.5 + 3.57i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.739 − 1.28i)3-s + (−0.249 + 0.433i)4-s + (−0.522 + 0.905i)6-s − 0.165·7-s + 0.353·8-s + (−0.593 + 1.02i)9-s + 0.301·11-s + 0.739·12-s + (−0.277 + 0.480i)13-s + (0.0585 + 0.101i)14-s + (−0.125 − 0.216i)16-s + (0.621 + 1.07i)17-s + 0.839·18-s + (−0.573 + 0.819i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.955 + 0.295i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.955 + 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.720631 - 0.108940i\)
\(L(\frac12)\) \(\approx\) \(0.720631 - 0.108940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
19 \( 1 + (2.5 - 3.57i)T \)
good3 \( 1 + (1.28 + 2.21i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 0.438T + 7T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.56 - 4.43i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.34 - 4.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 4.68T + 37T^{2} \)
41 \( 1 + (-3.06 - 5.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.56 - 2.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.43 + 2.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.78 - 6.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.28 + 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.56 - 4.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.71 + 8.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.12 + 14.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.842 + 1.45i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.56 - 9.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + (-1.34 + 2.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.842 - 1.45i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.08591534982631190882523816716, −9.291999136530214864891043819226, −8.034391012194054811197451773689, −7.71353917563814611499713700706, −6.40431852113881690202360225857, −6.09619932431761870005228223442, −4.67860278308229281341404030444, −3.47549000116529370054881429046, −2.03685003497668402038170409333, −1.16614741241938070408549416878, 0.49198785928873035586696560899, 2.79327854421939094898649892545, 4.22050970552036277765789174689, 4.82585311783983490856143275476, 5.73845687576166860524115295903, 6.50339896577056259295196510060, 7.54308161069149032378403271335, 8.548760555123878621234908043410, 9.449758665055224898959183682739, 9.979078343451387869442993613943

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