Properties

Label 2-950-19.4-c1-0-25
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: $\chi_{950} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.292 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80554 - 1.33519i\)
\(L(\frac12)\) \(\approx\) \(1.80554 - 1.33519i\)
\(L(\frac{3}{2})\) 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

−10.01438430129168742518728556540, −8.789292668895798696747022420537, −7.935545976279847340038355421649, −7.21989375687462602671257851068, −6.50778222849951419217814526987, −5.58994876997570259419281812323, −4.58926795963418343891665357897, −3.61219086437658350853062616139, −2.35848359323021166202161791842, −0.889889148906881428928260251101, 1.82342096416056628880341265834, 2.87989935964946855752428676172, 4.14048299653975111019713460927, 5.01571326403903683169524005703, 5.37491340544867495031055415808, 6.73077503719708727526305976350, 7.64004631740843724619822011253, 8.565711071187225817373030177883, 9.746464809701682961679889524720, 10.30201493453281553161735208170

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