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Properties

Label 2-950-19.5-c1-0-1
Degree $2$
Conductor $950$
Sign $0.292 - 0.956i$
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  + (−0.939 + 0.342i)2-s + (0.227 − 1.28i)3-s + (0.766 − 0.642i)4-s + (0.227 + 1.28i)6-s + (−1.11 − 1.93i)7-s + (−0.500 + 0.866i)8-s + (1.20 + 0.440i)9-s + (−2.90 + 5.03i)11-s + (−0.654 − 1.13i)12-s + (0.492 + 2.79i)13-s + (1.70 + 1.43i)14-s + (0.173 − 0.984i)16-s + (−1.00 + 0.366i)17-s − 1.28·18-s + (−2.13 + 3.80i)19-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.131 − 0.744i)3-s + (0.383 − 0.321i)4-s + (0.0927 + 0.526i)6-s + (−0.421 − 0.730i)7-s + (−0.176 + 0.306i)8-s + (0.403 + 0.146i)9-s + (−0.875 + 1.51i)11-s + (−0.188 − 0.327i)12-s + (0.136 + 0.774i)13-s + (0.456 + 0.383i)14-s + (0.0434 − 0.246i)16-s + (−0.244 + 0.0888i)17-s − 0.303·18-s + (−0.488 + 0.872i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.292 - 0.956i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.292 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652936 + 0.482843i\)
\(L(\frac12)\) \(\approx\) \(0.652936 + 0.482843i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 \)
19 \( 1 + (2.13 - 3.80i)T \)
good3 \( 1 + (-0.227 + 1.28i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (1.11 + 1.93i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.90 - 5.03i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.492 - 2.79i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.00 - 0.366i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (3.46 - 2.90i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.483 - 0.175i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.47 - 6.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.58T + 37T^{2} \)
41 \( 1 + (0.665 - 3.77i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-6.41 - 5.38i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (10.3 + 3.77i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-2.26 + 1.89i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (6.57 - 2.39i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-11.2 + 9.47i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (7.63 + 2.77i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-5.09 - 4.27i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.589 - 3.34i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.901 - 5.11i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-8.27 - 14.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.355 - 2.01i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (4.68 - 1.70i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.02298770537905512672990122558, −9.588029624162556798599667491593, −8.231394119028788264804164315399, −7.71779101247658962457241509922, −6.89895034481852644884797430116, −6.42984842424772088706383257567, −4.97159558771121946094883034322, −3.98454969842531413251468146216, −2.33565946442279374435983936115, −1.44367646284829779819289578333, 0.46921411130766217825519753104, 2.49926483999821346954632232245, 3.24927325645493161522905938311, 4.41479072493061124791543096231, 5.64155294835431073515507898483, 6.35361531924246842901466170805, 7.62066030341862809726044631120, 8.492730658649497731544395647792, 9.019714658448971031674082680640, 9.902259087331520233630912382762

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