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Properties

Label 2-950-95.64-c1-0-22
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: $\chi_{950} (349, \cdot )$
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

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

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