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Properties

Label 2-950-19.9-c1-0-18
Degree $2$
Conductor $950$
Sign $-0.249 + 0.968i$
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 + (−0.266 + 0.223i)3-s + (−0.939 + 0.342i)4-s + (0.266 + 0.223i)6-s + (−0.879 + 1.52i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 2.83i)9-s + (−2.11 − 3.66i)11-s + (0.173 − 0.300i)12-s + (−0.815 − 0.684i)13-s + (1.65 + 0.601i)14-s + (0.766 − 0.642i)16-s + (−1.23 − 7.02i)17-s + 2.87·18-s + (3.93 + 1.86i)19-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.153 + 0.128i)3-s + (−0.469 + 0.171i)4-s + (0.108 + 0.0911i)6-s + (−0.332 + 0.575i)7-s + (0.176 + 0.306i)8-s + (−0.166 + 0.945i)9-s + (−0.637 − 1.10i)11-s + (0.0501 − 0.0868i)12-s + (−0.226 − 0.189i)13-s + (0.441 + 0.160i)14-s + (0.191 − 0.160i)16-s + (−0.300 − 1.70i)17-s + 0.678·18-s + (0.903 + 0.427i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.249 + 0.968i$
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: \((2,\ 950,\ (\ :1/2),\ -0.249 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.599109 - 0.772764i\)
\(L(\frac12)\) \(\approx\) \(0.599109 - 0.772764i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.93 - 1.86i)T \)
good3 \( 1 + (0.266 - 0.223i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.879 - 1.52i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.11 + 3.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.815 + 0.684i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.23 + 7.02i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-3.53 + 1.28i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.10 + 6.27i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-4.41 + 7.64i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.45T + 37T^{2} \)
41 \( 1 + (-1.43 + 1.20i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.47 - 1.26i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.638 + 3.61i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (9.29 - 3.38i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.26 + 7.18i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (4.98 - 1.81i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.02 + 11.4i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.65 + 0.965i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.607 - 0.509i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (5.12 - 4.30i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.754 + 1.30i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-9.12 - 7.65i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.326 + 1.85i)T + (-91.1 + 33.1i)T^{2} \)
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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

−9.736640214291339363878474162197, −9.250554833521652045252078966724, −8.110830219623968079484077588691, −7.60665731181316993552018197817, −6.11700729785551719286628506065, −5.33257476467850174475494365827, −4.50724863679504146025895321486, −2.98184869771006224696806492310, −2.50504155224878997129816265035, −0.54264628082630990141575080278, 1.26160335600030764478292301919, 3.06868549300894233607376266192, 4.20765597589557008444352867702, 5.11748654597405523333899383910, 6.20001865060152231950980050766, 6.92575225227856375925708028160, 7.53746458877337481269892302612, 8.606733709633717881534188092930, 9.386682131303955361302267765180, 10.16071639783222873089312242019

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