Properties

Label 2-950-95.9-c1-0-29
Degree $2$
Conductor $950$
Sign $-0.939 + 0.341i$
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  + (0.984 − 0.173i)2-s + (−0.549 − 0.654i)3-s + (0.939 − 0.342i)4-s + (−0.654 − 0.549i)6-s + (−2.40 − 1.39i)7-s + (0.866 − 0.5i)8-s + (0.394 − 2.23i)9-s + (−1.58 − 2.74i)11-s + (−0.739 − 0.427i)12-s + (−3.08 + 3.68i)13-s + (−2.61 − 0.951i)14-s + (0.766 − 0.642i)16-s + (−1.33 + 0.235i)17-s − 2.27i·18-s + (−4.15 − 1.31i)19-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (−0.317 − 0.377i)3-s + (0.469 − 0.171i)4-s + (−0.267 − 0.224i)6-s + (−0.910 − 0.525i)7-s + (0.306 − 0.176i)8-s + (0.131 − 0.745i)9-s + (−0.477 − 0.826i)11-s + (−0.213 − 0.123i)12-s + (−0.856 + 1.02i)13-s + (−0.698 − 0.254i)14-s + (0.191 − 0.160i)16-s + (−0.324 + 0.0571i)17-s − 0.535i·18-s + (−0.953 − 0.300i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.939 + 0.341i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.939 + 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.182294 - 1.03575i\)
\(L(\frac12)\) \(\approx\) \(0.182294 - 1.03575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
5 \( 1 \)
19 \( 1 + (4.15 + 1.31i)T \)
good3 \( 1 + (0.549 + 0.654i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (2.40 + 1.39i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.58 + 2.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.08 - 3.68i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.33 - 0.235i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-2.06 - 5.66i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.81 + 10.2i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.80 - 6.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.9iT - 37T^{2} \)
41 \( 1 + (-2.86 + 2.40i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.525 - 1.44i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (2.92 + 0.516i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (1.47 + 4.04i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.714 + 4.04i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-3.98 + 1.44i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-5.63 - 0.992i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (0.844 + 0.307i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (6.72 + 8.01i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-1.71 + 1.43i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-7.21 - 4.16i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.26 - 6.09i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-2.58 + 0.456i)T + (91.1 - 33.1i)T^{2} \)
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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

−9.684058375506241987566256655021, −9.046118677040104381853787931854, −7.67984632698846735563342094680, −6.82582513198780574541908093908, −6.33691207759675700986137240248, −5.36517516582624698388049269515, −4.16692425930612553398721991375, −3.39719301620421694410919287836, −2.12282289956245517422827841595, −0.36962945151937114081140842386, 2.25662782470815404070617785426, 3.07287230671830576769162010774, 4.49909954501730337444035838530, 5.04687166111214872141678566778, 5.97545203251157643835261923015, 6.87879761392499687873481798251, 7.73956293010533672855476814821, 8.707044643347699235320029183345, 9.903587522834114863792398349955, 10.36039471212650618117143099437

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