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Properties

Label 2-950-19.11-c1-0-13
Degree $2$
Conductor $950$
Sign $0.910 + 0.412i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 7-s − 0.999·8-s + (1.5 + 2.59i)9-s + 5·11-s + (1 + 1.73i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 3·18-s + (−0.5 + 4.33i)19-s + (2.5 − 4.33i)22-s + (0.5 + 0.866i)23-s + 1.99·26-s + (0.499 + 0.866i)28-s + (−3 − 5.19i)29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.377·7-s − 0.353·8-s + (0.5 + 0.866i)9-s + 1.50·11-s + (0.277 + 0.480i)13-s + (−0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + 0.707·18-s + (−0.114 + 0.993i)19-s + (0.533 − 0.923i)22-s + (0.104 + 0.180i)23-s + 0.392·26-s + (0.0944 + 0.163i)28-s + (−0.557 − 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94884 - 0.421198i\)
\(L(\frac12)\) \(\approx\) \(1.94884 - 0.421198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
19 \( 1 + (0.5 - 4.33i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 11T + 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.932918427995837414089519391319, −9.430729454746527553327742374237, −8.447918354250738830884132284406, −7.40655059846664527146659403522, −6.41831436480062518115554336333, −5.64848032313846357543304691273, −4.29353267796000027187381777841, −3.87614472238841220322712760618, −2.40931606491552331823877825361, −1.29793947836325581910441911715, 1.04987745002552233066082512272, 2.98054166850019483021196192632, 3.92842817128213055915857811861, 4.73339480470200908725662950753, 6.11675240902167177274167002985, 6.51779602412075469500115003179, 7.35155301191210300128862824800, 8.433991680114459480987162768451, 9.317519993778239477508888710175, 9.719832933752611909378027284309

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