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Properties

Label 2-950-95.88-c1-0-4
Degree $2$
Conductor $950$
Sign $-0.288 - 0.957i$
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 + (1.61 + 0.431i)3-s + (−0.866 − 0.499i)4-s + (0.834 − 1.44i)6-s + (−2.81 + 2.81i)7-s + (−0.707 + 0.707i)8-s + (−0.185 − 0.107i)9-s − 5.57·11-s + (−1.18 − 1.18i)12-s + (0.831 + 3.10i)13-s + (1.98 + 3.44i)14-s + (0.500 + 0.866i)16-s + (−3.97 − 1.06i)17-s + (−0.151 + 0.151i)18-s + (0.254 + 4.35i)19-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.930 + 0.249i)3-s + (−0.433 − 0.249i)4-s + (0.340 − 0.590i)6-s + (−1.06 + 1.06i)7-s + (−0.249 + 0.249i)8-s + (−0.0618 − 0.0356i)9-s − 1.68·11-s + (−0.340 − 0.340i)12-s + (0.230 + 0.861i)13-s + (0.531 + 0.919i)14-s + (0.125 + 0.216i)16-s + (−0.963 − 0.258i)17-s + (−0.0356 + 0.0356i)18-s + (0.0584 + 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.288 - 0.957i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.288 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.468543 + 0.630843i\)
\(L(\frac12)\) \(\approx\) \(0.468543 + 0.630843i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.254 - 4.35i)T \)
good3 \( 1 + (-1.61 - 0.431i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (2.81 - 2.81i)T - 7iT^{2} \)
11 \( 1 + 5.57T + 11T^{2} \)
13 \( 1 + (-0.831 - 3.10i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.97 + 1.06i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (-3.83 + 1.02i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.09 - 3.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.573iT - 31T^{2} \)
37 \( 1 + (-7.24 - 7.24i)T + 37iT^{2} \)
41 \( 1 + (5.12 - 2.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.44 + 5.37i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.15 + 11.7i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.981 - 3.66i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.75 + 6.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.06 - 1.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.4 + 3.07i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (4.56 - 2.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.50 - 13.0i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.48 + 4.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.30 - 7.30i)T + 83iT^{2} \)
89 \( 1 + (7.08 - 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.619 + 2.31i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.08090860276530474717611881447, −9.506642830078638981783691364461, −8.734397262401046253913455988205, −8.219578275678427799069762723515, −6.84556221344604762127557863456, −5.83371107048071321600530249961, −4.92365712710112897398538156879, −3.64075276449251023802110426161, −2.84762386434567724457334907398, −2.19006918227792229726334365949, 0.27859483097321095841560288552, 2.60392810155176097980780490674, 3.26422433568934044689286402701, 4.44549812719164819317681087835, 5.52377852098656903020831379673, 6.51011818922695733196041394742, 7.53841182918860403501049642077, 7.79759678296802774616840037224, 8.845391181092235735501299074991, 9.577201082988873278855360473738

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