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Properties

Label 2-950-95.49-c1-0-19
Degree $2$
Conductor $950$
Sign $-0.575 + 0.817i$
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 + (−1.47 − 0.851i)3-s + (0.499 + 0.866i)4-s + (0.851 + 1.47i)6-s − 3.74i·7-s − 0.999i·8-s + (−0.0492 − 0.0852i)9-s + 3.64·11-s − 1.70i·12-s + (5.23 − 3.01i)13-s + (−1.87 + 3.24i)14-s + (−0.5 + 0.866i)16-s + (3.54 + 2.04i)17-s + 0.0984i·18-s + (0.697 + 4.30i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.851 − 0.491i)3-s + (0.249 + 0.433i)4-s + (0.347 + 0.602i)6-s − 1.41i·7-s − 0.353i·8-s + (−0.0164 − 0.0284i)9-s + 1.09·11-s − 0.491i·12-s + (1.45 − 0.837i)13-s + (−0.500 + 0.866i)14-s + (−0.125 + 0.216i)16-s + (0.860 + 0.497i)17-s + 0.0231i·18-s + (0.160 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.575 + 0.817i$
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: \((2,\ 950,\ (\ :1/2),\ -0.575 + 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.426414 - 0.821962i\)
\(L(\frac12)\) \(\approx\) \(0.426414 - 0.821962i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.697 - 4.30i)T \)
good3 \( 1 + (1.47 + 0.851i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.74iT - 7T^{2} \)
11 \( 1 - 3.64T + 11T^{2} \)
13 \( 1 + (-5.23 + 3.01i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.54 - 2.04i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.05 - 2.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.32 + 5.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 + 7.75iT - 37T^{2} \)
41 \( 1 + (-3.99 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.80 + 2.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.50 + 0.871i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.50 + 0.871i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.67 - 6.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.15 + 1.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.83 - 3.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.994 + 1.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.95 + 4.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.07 + 5.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.69iT - 83T^{2} \)
89 \( 1 + (5.53 + 9.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.30 - 0.752i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.08813021749990395065873646594, −8.932786155201019152056665882194, −7.953102674501868814722920438061, −7.34351506436602177716673338366, −6.22123941718303106320315540399, −5.86112441808051723485916493111, −4.01298717817309907462887713989, −3.55063028832187021057856011319, −1.45244346206368166916246875120, −0.72826884029468695304364096955, 1.35248002451110378295912532316, 2.86678153780234802492758384624, 4.38527203018845073010405083734, 5.29327823177135352619056748076, 6.22148005320612470960297412355, 6.54384746721397863941672572589, 8.063196181559678405937424836188, 8.774573983994419276884984363751, 9.399968394658265215352467678711, 10.21407558822704510524853960258

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