Properties

Label 2-950-19.5-c1-0-11
Degree $2$
Conductor $950$
Sign $0.992 - 0.120i$
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 + (0.0603 − 0.342i)3-s + (0.766 − 0.642i)4-s + (−0.0603 − 0.342i)6-s + (0.766 + 1.32i)7-s + (0.500 − 0.866i)8-s + (2.70 + 0.984i)9-s + (−1.55 + 2.68i)11-s + (−0.173 − 0.300i)12-s + (0.794 + 4.50i)13-s + (1.17 + 0.984i)14-s + (0.173 − 0.984i)16-s + (−1.99 + 0.725i)17-s + 2.87·18-s + (4.34 + 0.405i)19-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.0348 − 0.197i)3-s + (0.383 − 0.321i)4-s + (−0.0246 − 0.139i)6-s + (0.289 + 0.501i)7-s + (0.176 − 0.306i)8-s + (0.901 + 0.328i)9-s + (−0.468 + 0.811i)11-s + (−0.0501 − 0.0868i)12-s + (0.220 + 1.24i)13-s + (0.313 + 0.263i)14-s + (0.0434 − 0.246i)16-s + (−0.483 + 0.175i)17-s + 0.678·18-s + (0.995 + 0.0929i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.992 - 0.120i$
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: \((2,\ 950,\ (\ :1/2),\ 0.992 - 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58574 + 0.155859i\)
\(L(\frac12)\) \(\approx\) \(2.58574 + 0.155859i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.34 - 0.405i)T \)
good3 \( 1 + (-0.0603 + 0.342i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (-0.766 - 1.32i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.55 - 2.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.794 - 4.50i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.99 - 0.725i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.25 + 1.89i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-6.12 - 2.22i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.29 + 5.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.45T + 37T^{2} \)
41 \( 1 + (-0.773 + 4.38i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (2.20 + 1.85i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-3.25 - 1.18i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (0.124 - 0.104i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-13.3 + 4.84i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (6.14 - 5.15i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (8.07 + 2.94i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-7.37 - 6.18i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.538 - 3.05i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.173 - 0.984i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (2.66 + 4.61i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.21 - 6.91i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (5.77 - 2.10i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.17149049747385242195646487050, −9.327358000503599205016561445186, −8.387958846700883529874801154924, −7.19585825632612274461771969808, −6.81182236992194436895871299972, −5.50507579999601021680326872207, −4.72872911849213580370538934219, −3.90961813779821085817641056233, −2.44580249717101657614274819911, −1.61260937413580902929413905316, 1.12057135072876904889104095088, 2.93904173968376261911089112668, 3.69357638218139048366660404193, 4.83798062771261376120918357930, 5.49373225772977861423087889025, 6.62238665256785314212043744289, 7.41002510303399399688637079351, 8.174375309472752119046812549964, 9.144291137004098053237233731773, 10.32345551005120947821154543168

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