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Properties

Label 2-950-19.11-c1-0-17
Degree $2$
Conductor $950$
Sign $0.813 - 0.582i$
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  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 4·7-s + 0.999·8-s + (1 + 1.73i)9-s + 3·11-s − 0.999·12-s + (1 + 1.73i)13-s + (−2 + 3.46i)14-s + (−0.5 + 0.866i)16-s + (−3 + 5.19i)17-s − 2·18-s + (−3.5 − 2.59i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.204 + 0.353i)6-s + 1.51·7-s + 0.353·8-s + (0.333 + 0.577i)9-s + 0.904·11-s − 0.288·12-s + (0.277 + 0.480i)13-s + (−0.534 + 0.925i)14-s + (−0.125 + 0.216i)16-s + (−0.727 + 1.26i)17-s − 0.471·18-s + (−0.802 − 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.813 - 0.582i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.813 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70040 + 0.546084i\)
\(L(\frac12)\) \(\approx\) \(1.70040 + 0.546084i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
19 \( 1 + (3.5 + 2.59i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.09474199711844011687907176169, −8.874323509406739257606047355231, −8.345774465162069181527750648356, −7.81236668075384802346719820142, −6.72485337033489874047842306545, −6.15593470953801047883776310921, −4.65934366186294047675663120380, −4.30568778485884231597679763653, −2.19508762073766577431119049007, −1.39598196030327509443736632745, 1.12913565088584557117681843960, 2.30369861584046849330064066884, 3.73474008897104265396212364368, 4.33593392557876322772710668377, 5.36774904382666829579776527488, 6.68664673243537788932100393543, 7.69992835747480889570578386681, 8.519249016691948735719943177521, 9.149570185205668476990639362047, 9.946850356494535096082866129794

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