Properties

Label 2-950-95.49-c1-0-17
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 + (2.82 + 1.63i)3-s + (0.499 + 0.866i)4-s + (−1.63 − 2.82i)6-s + 2.62i·7-s − 0.999i·8-s + (3.82 + 6.63i)9-s + 5.03·11-s + 3.26i·12-s + (4.02 − 2.32i)13-s + (1.31 − 2.26i)14-s + (−0.5 + 0.866i)16-s + (−3.16 − 1.82i)17-s − 7.65i·18-s + (−0.697 − 4.30i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (1.63 + 0.942i)3-s + (0.249 + 0.433i)4-s + (−0.666 − 1.15i)6-s + 0.990i·7-s − 0.353i·8-s + (1.27 + 2.21i)9-s + 1.51·11-s + 0.942i·12-s + (1.11 − 0.644i)13-s + (0.350 − 0.606i)14-s + (−0.125 + 0.216i)16-s + (−0.768 − 0.443i)17-s − 1.80i·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

\(L(1)\) \(\approx\) \(2.01366 + 1.04464i\)
\(L(\frac12)\) \(\approx\) \(2.01366 + 1.04464i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.82 - 1.63i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 2.62iT - 7T^{2} \)
11 \( 1 - 5.03T + 11T^{2} \)
13 \( 1 + (-4.02 + 2.32i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.16 + 1.82i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.05 - 2.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.01 + 6.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 - 5.75iT - 37T^{2} \)
41 \( 1 + (-2.90 + 5.03i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.502 - 0.290i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.00 - 2.31i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.00 - 2.31i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.88 - 3.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0650 + 0.112i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.328 - 0.189i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.56 + 7.91i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.36 - 4.25i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.98 - 6.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.86iT - 83T^{2} \)
89 \( 1 + (4.14 + 7.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.0 + 8.09i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.604380918077516384224015172991, −9.437347452984636233548349461332, −8.633137254529213680122288964632, −8.219710375847734131268433811078, −7.08499214207683716460010871714, −5.86875483569214919125791115554, −4.41083205277896193825377183481, −3.66786088921721146186088158248, −2.73940693142382977869797523683, −1.78561566062354957624069080558, 1.27897762587160084145662048947, 1.92117771926075669215304282182, 3.66378849561779611757441135274, 4.01651401488423205660171820622, 6.27968005537183651296640530993, 6.69249008457681516189699262020, 7.51364839463127636469934927270, 8.282116179547977542650545608960, 8.958209828625179347383071748743, 9.444256798330978607045711288596

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