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Properties

Label 2-950-19.11-c1-0-3
Degree $2$
Conductor $950$
Sign $-0.604 - 0.796i$
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 + (0.780 − 1.35i)3-s + (−0.499 − 0.866i)4-s + (0.780 + 1.35i)6-s − 4.56·7-s + 0.999·8-s + (0.280 + 0.486i)9-s + 11-s − 1.56·12-s + (−1 − 1.73i)13-s + (2.28 − 3.95i)14-s + (−0.5 + 0.866i)16-s + (−1.56 + 2.70i)17-s − 0.561·18-s + (−2.5 + 3.57i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.450 − 0.780i)3-s + (−0.249 − 0.433i)4-s + (0.318 + 0.552i)6-s − 1.72·7-s + 0.353·8-s + (0.0935 + 0.162i)9-s + 0.301·11-s − 0.450·12-s + (−0.277 − 0.480i)13-s + (0.609 − 1.05i)14-s + (−0.125 + 0.216i)16-s + (−0.378 + 0.655i)17-s − 0.132·18-s + (−0.573 + 0.819i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.604 - 0.796i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.265334 + 0.534491i\)
\(L(\frac12)\) \(\approx\) \(0.265334 + 0.534491i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.780 + 1.35i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 4.56T + 7T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.56 - 2.70i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.84 - 6.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.24T + 31T^{2} \)
37 \( 1 + 7.68T + 37T^{2} \)
41 \( 1 + (1.06 - 1.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.56 - 4.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.56 - 9.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.71 + 2.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.21 - 9.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.56 - 2.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.78 - 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.123 + 0.213i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.34 + 9.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.43 + 2.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.80T + 83T^{2} \)
89 \( 1 + (4.84 + 8.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.34 - 9.25i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.10024880455393892629858918594, −9.369700001476281435010347671809, −8.621496417074650732662705969274, −7.66121740051654798537837541140, −7.03647425469056582704782230544, −6.30171102648632785069654370486, −5.48152781005895761180796559248, −3.96314801014182569624082848351, −2.92781513086333304502025845075, −1.54026414417845786939621825360, 0.29114173388607204644534243293, 2.39325407438640905907261968203, 3.32582065808810436017132429337, 4.01972636108861591190358693220, 5.09561205512416389199311598435, 6.77053026046541362276791794488, 6.87079552181877487720511514253, 8.661604180247012200214264859443, 9.111131635142693912971490433463, 9.640587671218529225703441010320

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