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Properties

Label 2-950-19.9-c1-0-26
Degree $2$
Conductor $950$
Sign $-0.600 + 0.799i$
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 + (0.639 − 0.536i)3-s + (−0.939 + 0.342i)4-s + (0.639 + 0.536i)6-s + (−2.09 + 3.62i)7-s + (−0.5 − 0.866i)8-s + (−0.399 + 2.26i)9-s + (−2.69 − 4.66i)11-s + (−0.417 + 0.723i)12-s + (−3.01 − 2.52i)13-s + (−3.93 − 1.43i)14-s + (0.766 − 0.642i)16-s + (−0.371 − 2.10i)17-s − 2.30·18-s + (−2.92 − 3.22i)19-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (0.369 − 0.309i)3-s + (−0.469 + 0.171i)4-s + (0.261 + 0.219i)6-s + (−0.790 + 1.36i)7-s + (−0.176 − 0.306i)8-s + (−0.133 + 0.755i)9-s + (−0.812 − 1.40i)11-s + (−0.120 + 0.208i)12-s + (−0.835 − 0.700i)13-s + (−1.05 − 0.382i)14-s + (0.191 − 0.160i)16-s + (−0.0901 − 0.511i)17-s − 0.542·18-s + (−0.671 − 0.740i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.600 + 0.799i$
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: \((2,\ 950,\ (\ :1/2),\ -0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00852454 - 0.0170635i\)
\(L(\frac12)\) \(\approx\) \(0.00852454 - 0.0170635i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.92 + 3.22i)T \)
good3 \( 1 + (-0.639 + 0.536i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (2.09 - 3.62i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.69 + 4.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.01 + 2.52i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.371 + 2.10i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (2.68 - 0.976i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.34 + 7.65i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.86 - 4.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.84T + 37T^{2} \)
41 \( 1 + (-4.67 + 3.92i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-11.8 - 4.31i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.760 - 4.31i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (12.2 - 4.44i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.36 + 7.75i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (10.3 - 3.74i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.82 - 10.3i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (12.6 + 4.61i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (2.58 - 2.16i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (3.96 - 3.32i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.36 - 7.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.52 + 2.12i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.150 + 0.851i)T + (-91.1 + 33.1i)T^{2} \)
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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

−9.428996683917092564310939242672, −8.764734998187671989770702406506, −8.042091421777243419339187246637, −7.36471612239840492128296567126, −6.06326917678339613443446365705, −5.67324819372398998759465307738, −4.68582113579226315312652992530, −2.97141492050391098443332227247, −2.56301749397654926013617614009, −0.00750620324238927431978635350, 1.84725571382805038881689251887, 3.07992101966791788746976365802, 4.10442410043760956906037829510, 4.54911437437621161433013109095, 6.06935190449858768569495339311, 7.05316484628304408778139839327, 7.75683774909024877962513529023, 9.034319019762373851003561329714, 9.709545963930151712713242900095, 10.26483823353879596973927629654

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