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Properties

Label 2-950-19.5-c1-0-0
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: $\chi_{950} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.838 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153884 - 0.518445i\)
\(L(\frac12)\) \(\approx\) \(0.153884 - 0.518445i\)
\(L(\frac{3}{2})\) 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

−10.22283435099180943326047463601, −9.843117011589328151228221585606, −8.887614594290209091269702921058, −8.429045096488369640891139684855, −7.17185547314669685455789195736, −6.17025996641242538916131586215, −5.34864803021569621394930212868, −4.35329338567081476805718018993, −3.49528574640489164235446527731, −1.90574733139794780147523018491, 0.31610712765525937661583068366, 1.60407456994380498821118910710, 2.53802602932171175678443505779, 3.97484358766073781620668590474, 5.46477395327851422381486057082, 6.43111721943969860891133719310, 7.09652084584255814267015038263, 7.85720508571234999322177445024, 8.500334026123495405124481072113, 9.483087611221734436169645172563

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