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Properties

Label 2-950-95.49-c1-0-4
Degree $2$
Conductor $950$
Sign $-0.157 - 0.987i$
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  + (0.866 + 0.5i)2-s + (−1.55 − 0.895i)3-s + (0.499 + 0.866i)4-s + (−0.895 − 1.55i)6-s + i·7-s + 0.999i·8-s + (0.104 + 0.180i)9-s + 0.791·11-s − 1.79i·12-s + (−4.14 + 2.39i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (3.28 + 1.89i)17-s + 0.208i·18-s + (3.5 + 2.59i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.895 − 0.517i)3-s + (0.249 + 0.433i)4-s + (−0.365 − 0.633i)6-s + 0.377i·7-s + 0.353i·8-s + (0.0347 + 0.0602i)9-s + 0.238·11-s − 0.517i·12-s + (−1.15 + 0.664i)13-s + (−0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.796 + 0.459i)17-s + 0.0491i·18-s + (0.802 + 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.157 - 0.987i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.157 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794753 + 0.931257i\)
\(L(\frac12)\) \(\approx\) \(0.794753 + 0.931257i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
19 \( 1 + (-3.5 - 2.59i)T \)
good3 \( 1 + (1.55 + 0.895i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 - 0.791T + 11T^{2} \)
13 \( 1 + (4.14 - 2.39i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.28 - 1.89i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.96 - 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.10 - 1.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.208T + 31T^{2} \)
37 \( 1 - 5.58iT - 37T^{2} \)
41 \( 1 + (1.18 - 2.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.04 + 0.604i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.33 - 3.08i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.96 + 2.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.29 - 3.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.18 - 10.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.55 - 3.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.29 - 3.96i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.55 - 0.895i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.97 - 6.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.791iT - 83T^{2} \)
89 \( 1 + (2.29 + 3.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.18 - 0.686i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.29521806492767286587658411084, −9.520275644993420033002442537231, −8.398488648341513825491179321167, −7.42934720322410379059236550176, −6.77555182433010617708373862331, −5.83556462939234929711176604288, −5.34599946740540913040218219948, −4.20459986011260021145760166174, −3.01024146138944610119644085571, −1.53089659550587831425513664400, 0.52931753622121281085655738764, 2.38957868283398017958556354883, 3.56662037305859099462501661891, 4.69103687700323515949380427193, 5.23964621876271365325552938228, 6.05906362692532813098942433114, 7.15431770609524371949146575464, 7.952484728322151347773318884519, 9.375547047224392170766567534291, 10.12085992484952293014644023378

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