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Properties

Label 2-950-19.4-c1-0-9
Degree $2$
Conductor $950$
Sign $-0.838 - 0.545i$
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 + (0.439 + 2.49i)3-s + (0.766 + 0.642i)4-s + (−0.439 + 2.49i)6-s + (−0.326 + 0.565i)7-s + (0.500 + 0.866i)8-s + (−3.20 + 1.16i)9-s + (0.5 + 0.866i)11-s + (−1.26 + 2.19i)12-s + (−0.5 + 2.83i)13-s + (−0.5 + 0.419i)14-s + (0.173 + 0.984i)16-s + (0.439 + 0.160i)17-s − 3.41·18-s + (−4.07 − 1.55i)19-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.253 + 1.43i)3-s + (0.383 + 0.321i)4-s + (−0.179 + 1.01i)6-s + (−0.123 + 0.213i)7-s + (0.176 + 0.306i)8-s + (−1.06 + 0.388i)9-s + (0.150 + 0.261i)11-s + (−0.365 + 0.633i)12-s + (−0.138 + 0.786i)13-s + (−0.133 + 0.112i)14-s + (0.0434 + 0.246i)16-s + (0.106 + 0.0388i)17-s − 0.804·18-s + (−0.934 − 0.355i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.838 - 0.545i$
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: \((2,\ 950,\ (\ :1/2),\ -0.838 - 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.664867 + 2.23997i\)
\(L(\frac12)\) \(\approx\) \(0.664867 + 2.23997i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.07 + 1.55i)T \)
good3 \( 1 + (-0.439 - 2.49i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (0.326 - 0.565i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 2.83i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.439 - 0.160i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-2.56 - 2.15i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (6.41 - 2.33i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.03 + 3.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.63T + 37T^{2} \)
41 \( 1 + (0.854 + 4.84i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (2.40 - 2.02i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-6.02 + 2.19i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (2.96 + 2.48i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-5.68 - 2.06i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (3.96 + 3.32i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (0.826 - 0.300i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-9.35 + 7.84i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.0885 + 0.502i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.51 + 8.58i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-8.95 + 15.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.06 - 17.3i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-13.6 - 4.96i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.43586896888432346191633341941, −9.350305863342693195294266462746, −9.077921553898979911152385708487, −7.87439930260833248207740154945, −6.84710100974077050969059368514, −5.86623334228308907396135096077, −4.89610199954287676935309577419, −4.22709543072097420660254242717, −3.43921966784764158192626973328, −2.23707688591666254237885515290, 0.856265623157924132484651765759, 2.11505230238034545182611705298, 3.05190313532651448053732997255, 4.23284456754973742710242467629, 5.52024215346842317463496907493, 6.32435969300490636212981619621, 7.05741559278710345993269960762, 7.87530242834371769695989308526, 8.599475964210350279735732038361, 9.809592694483676053257107886358

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