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Properties

Label 2-950-19.9-c1-0-10
Degree $2$
Conductor $950$
Sign $0.102 - 0.994i$
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  + (0.173 + 0.984i)2-s + (1.77 − 1.49i)3-s + (−0.939 + 0.342i)4-s + (1.77 + 1.49i)6-s + (−2.45 + 4.25i)7-s + (−0.5 − 0.866i)8-s + (0.413 − 2.34i)9-s + (−1.42 − 2.46i)11-s + (−1.15 + 2.00i)12-s + (5.02 + 4.21i)13-s + (−4.62 − 1.68i)14-s + (0.766 − 0.642i)16-s + (0.380 + 2.15i)17-s + 2.38·18-s + (4.17 + 1.26i)19-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (1.02 − 0.860i)3-s + (−0.469 + 0.171i)4-s + (0.725 + 0.608i)6-s + (−0.929 + 1.60i)7-s + (−0.176 − 0.306i)8-s + (0.137 − 0.781i)9-s + (−0.429 − 0.743i)11-s + (−0.334 + 0.579i)12-s + (1.39 + 1.16i)13-s + (−1.23 − 0.449i)14-s + (0.191 − 0.160i)16-s + (0.0922 + 0.523i)17-s + 0.561·18-s + (0.957 + 0.289i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.102 - 0.994i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.102 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46612 + 1.32239i\)
\(L(\frac12)\) \(\approx\) \(1.46612 + 1.32239i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 \)
19 \( 1 + (-4.17 - 1.26i)T \)
good3 \( 1 + (-1.77 + 1.49i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (2.45 - 4.25i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.42 + 2.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.02 - 4.21i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.380 - 2.15i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (2.49 - 0.908i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.32 - 7.48i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.70 + 4.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.06T + 37T^{2} \)
41 \( 1 + (7.16 - 6.01i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.62 - 0.593i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.521 + 2.95i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (3.96 - 1.44i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.167 + 0.949i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-7.04 + 2.56i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.333 - 1.89i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.79 - 2.10i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-3.04 + 2.55i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-11.7 + 9.84i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.656 - 1.13i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.25 + 6.92i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.356 + 2.02i)T + (-91.1 + 33.1i)T^{2} \)
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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

−9.769377365520336752136921966459, −9.007828043839125497987150394429, −8.523541812587874577154274704125, −7.903900763759290316979338902722, −6.71557571491608492132755637148, −6.15380695482972703704737269880, −5.35236832194951992158764026204, −3.64787271258327670313476028042, −2.94224588932224397088281328149, −1.69878327071180939271443799331, 0.825859487146942789671140672112, 2.69792212733985255431038372763, 3.55003171143091243022040697938, 4.00256072357720703107433704162, 5.10970276472220376508432418587, 6.44794572750888176390384954469, 7.58216544884151028599969843647, 8.282843429370674960290716652317, 9.369687308225238947759533983994, 9.977934215140521523437268203007

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