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Properties

Label 2-950-19.4-c1-0-18
Degree $2$
Conductor $950$
Sign $0.703 - 0.710i$
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 + (0.111 + 0.634i)3-s + (0.766 + 0.642i)4-s + (−0.111 + 0.634i)6-s + (0.213 − 0.369i)7-s + (0.500 + 0.866i)8-s + (2.42 − 0.883i)9-s + (1.50 + 2.60i)11-s + (−0.322 + 0.558i)12-s + (0.831 − 4.71i)13-s + (0.327 − 0.274i)14-s + (0.173 + 0.984i)16-s + (4.46 + 1.62i)17-s + 2.58·18-s + (−2.34 + 3.67i)19-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.0646 + 0.366i)3-s + (0.383 + 0.321i)4-s + (−0.0457 + 0.259i)6-s + (0.0807 − 0.139i)7-s + (0.176 + 0.306i)8-s + (0.809 − 0.294i)9-s + (0.453 + 0.786i)11-s + (−0.0930 + 0.161i)12-s + (0.230 − 1.30i)13-s + (0.0874 − 0.0733i)14-s + (0.0434 + 0.246i)16-s + (1.08 + 0.394i)17-s + 0.609·18-s + (−0.537 + 0.843i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.703 - 0.710i$
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: \((2,\ 950,\ (\ :1/2),\ 0.703 - 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48389 + 1.03527i\)
\(L(\frac12)\) \(\approx\) \(2.48389 + 1.03527i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.34 - 3.67i)T \)
good3 \( 1 + (-0.111 - 0.634i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-0.213 + 0.369i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.50 - 2.60i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.831 + 4.71i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-4.46 - 1.62i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (4.00 + 3.36i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-0.536 + 0.195i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-0.113 + 0.196i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.76T + 37T^{2} \)
41 \( 1 + (-1.60 - 9.12i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.777 - 0.652i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (6.78 - 2.46i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (8.29 + 6.95i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.330 + 0.120i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.0978 - 0.0820i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.246 + 0.0895i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-0.343 + 0.287i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.48 + 8.41i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.31 + 13.1i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.550 - 0.954i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.75 - 9.98i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (14.5 + 5.28i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.10039018374642286682842182436, −9.583084077616077463857866769553, −8.123703002572718525998536253512, −7.74304159591417148013205960223, −6.52973601685509763378137033452, −5.87034796403369314871574479885, −4.69122449728069156261044909996, −4.02092039216568271776280878345, −3.04224170039827725742035317347, −1.47416822421050170610496706324, 1.26991704395191587407215376977, 2.38563338902778929397735576691, 3.70788181120996618548870909803, 4.48155214063335895923401961878, 5.59176961049090277928739621331, 6.49146005190605034142514602828, 7.21778223422410379395569718715, 8.174069303688503483640853996747, 9.223837348523701652577275201165, 9.947447296056669075420510473419

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