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Properties

Label 2-950-95.27-c1-0-25
Degree $2$
Conductor $950$
Sign $-0.617 - 0.786i$
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  + (−0.258 − 0.965i)2-s + (−3.07 + 0.824i)3-s + (−0.866 + 0.499i)4-s + (1.59 + 2.75i)6-s + (−3.29 − 3.29i)7-s + (0.707 + 0.707i)8-s + (6.18 − 3.56i)9-s + 1.07·11-s + (2.25 − 2.25i)12-s + (0.265 − 0.991i)13-s + (−2.32 + 4.03i)14-s + (0.500 − 0.866i)16-s + (6.05 − 1.62i)17-s + (−5.04 − 5.04i)18-s + (−3.41 − 2.70i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−1.77 + 0.475i)3-s + (−0.433 + 0.249i)4-s + (0.649 + 1.12i)6-s + (−1.24 − 1.24i)7-s + (0.249 + 0.249i)8-s + (2.06 − 1.18i)9-s + 0.323·11-s + (0.649 − 0.649i)12-s + (0.0737 − 0.275i)13-s + (−0.622 + 1.07i)14-s + (0.125 − 0.216i)16-s + (1.46 − 0.393i)17-s + (−1.18 − 1.18i)18-s + (−0.783 − 0.620i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.617 - 0.786i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0473819 + 0.0975142i\)
\(L(\frac12)\) \(\approx\) \(0.0473819 + 0.0975142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
19 \( 1 + (3.41 + 2.70i)T \)
good3 \( 1 + (3.07 - 0.824i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (3.29 + 3.29i)T + 7iT^{2} \)
11 \( 1 - 1.07T + 11T^{2} \)
13 \( 1 + (-0.265 + 0.991i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-6.05 + 1.62i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-2.39 - 0.642i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.98 + 5.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.124iT - 31T^{2} \)
37 \( 1 + (3.61 - 3.61i)T - 37iT^{2} \)
41 \( 1 + (4.79 + 2.76i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.75 + 10.2i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.54 - 9.50i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.463 - 1.72i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.41 - 9.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.14 + 5.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.4 + 3.60i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.49 - 5.48i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.56 - 9.56i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.35 + 5.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.38 + 3.38i)T - 83iT^{2} \)
89 \( 1 + (2.05 + 3.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.48 - 9.27i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.917808794243426908021828374289, −9.186258804762303723068551308208, −7.56773989480168881987412111857, −6.79203713604548467811586324457, −6.01414421806745823901780271462, −5.01740078249196988133311713363, −4.07725166922047926666230794410, −3.30779317850132391995170348312, −1.07660093209611062300026026670, −0.085675364058506641853527951056, 1.59463143765275195312565874111, 3.50485459601160708041043730306, 4.98204896518084264914085746134, 5.65925914670103537924225915013, 6.31558030690184672344556524244, 6.76688853976409228391564792588, 7.84808310992524580224801165321, 8.945368456768456546601326292717, 9.828896464469790766244701230602, 10.48245463774449756018188301585

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