Properties

Label 2-950-95.24-c1-0-9
Degree $2$
Conductor $950$
Sign $-0.614 - 0.788i$
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  + (0.342 + 0.939i)2-s + (2.25 + 0.397i)3-s + (−0.766 + 0.642i)4-s + (0.397 + 2.25i)6-s + (−2.39 + 1.38i)7-s + (−0.866 − 0.500i)8-s + (2.10 + 0.764i)9-s + (−1.21 + 2.09i)11-s + (−1.98 + 1.14i)12-s + (1.70 − 0.300i)13-s + (−2.12 − 1.77i)14-s + (0.173 − 0.984i)16-s + (1.38 + 3.79i)17-s + 2.23i·18-s + (−0.0664 + 4.35i)19-s + ⋯
L(s)  = 1  + (0.241 + 0.664i)2-s + (1.30 + 0.229i)3-s + (−0.383 + 0.321i)4-s + (0.162 + 0.920i)6-s + (−0.906 + 0.523i)7-s + (−0.306 − 0.176i)8-s + (0.700 + 0.254i)9-s + (−0.365 + 0.632i)11-s + (−0.572 + 0.330i)12-s + (0.472 − 0.0833i)13-s + (−0.566 − 0.475i)14-s + (0.0434 − 0.246i)16-s + (0.334 + 0.919i)17-s + 0.527i·18-s + (−0.0152 + 0.999i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.614 - 0.788i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.614 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967424 + 1.98019i\)
\(L(\frac12)\) \(\approx\) \(0.967424 + 1.98019i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 - 0.939i)T \)
5 \( 1 \)
19 \( 1 + (0.0664 - 4.35i)T \)
good3 \( 1 + (-2.25 - 0.397i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (2.39 - 1.38i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.21 - 2.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.70 + 0.300i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.38 - 3.79i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-5.74 - 6.84i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (4.18 + 1.52i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.953 - 1.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.89iT - 37T^{2} \)
41 \( 1 + (0.808 - 4.58i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.91 + 2.28i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-3.39 + 9.33i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (7.18 + 8.56i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-1.89 + 0.689i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-6.66 + 5.59i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-4.61 + 12.6i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-10.2 - 8.57i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-5.62 - 0.991i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.16 - 6.62i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-14.6 + 8.47i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.964 + 5.46i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-2.63 - 7.22i)T + (-74.3 + 62.3i)T^{2} \)
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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

−9.770640397108549256620677245326, −9.521912135136195883091839300802, −8.524528592777577589678294187936, −7.967496489368814919745922044735, −7.06833044381450770437917050033, −6.04828009618461869306934889252, −5.20746380668361906843142543271, −3.72944707134270429461148335947, −3.35935392667852767861410418420, −2.03073137192104623407562468810, 0.819811112208797898434441412247, 2.57133285675857838917808516294, 3.05324620881851238797022556990, 3.98648541135256075987222859320, 5.16224773161116839363474128737, 6.44541516922020786543263519424, 7.29538482618196360050341665841, 8.265432479500359324841608008199, 9.110011846178804106857359502745, 9.507392279542907889464457115614

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