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Properties

Label 2-950-19.9-c1-0-14
Degree $2$
Conductor $950$
Sign $0.877 + 0.479i$
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 + (1.17 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−1.17 − 0.984i)6-s + (−0.939 + 1.62i)7-s + (0.5 + 0.866i)8-s + (−0.113 + 0.642i)9-s + (0.0812 + 0.140i)11-s + (−0.766 + 1.32i)12-s + (3.61 + 3.03i)13-s + (1.76 + 0.642i)14-s + (0.766 − 0.642i)16-s + (0.754 + 4.28i)17-s + 0.652·18-s + (2.77 − 3.35i)19-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (0.677 − 0.568i)3-s + (−0.469 + 0.171i)4-s + (−0.479 − 0.402i)6-s + (−0.355 + 0.615i)7-s + (0.176 + 0.306i)8-s + (−0.0377 + 0.214i)9-s + (0.0244 + 0.0424i)11-s + (−0.221 + 0.383i)12-s + (1.00 + 0.840i)13-s + (0.471 + 0.171i)14-s + (0.191 − 0.160i)16-s + (0.183 + 1.03i)17-s + 0.153·18-s + (0.637 − 0.770i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.877 + 0.479i$
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: \((2,\ 950,\ (\ :1/2),\ 0.877 + 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72902 - 0.441889i\)
\(L(\frac12)\) \(\approx\) \(1.72902 - 0.441889i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.77 + 3.35i)T \)
good3 \( 1 + (-1.17 + 0.984i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.939 - 1.62i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0812 - 0.140i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.61 - 3.03i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.754 - 4.28i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-5.73 + 2.08i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.414 + 2.35i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.81 + 3.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.36T + 37T^{2} \)
41 \( 1 + (-1.81 + 1.52i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.613 - 0.223i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.19 - 6.77i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-5.58 + 2.03i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.708 + 4.01i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.21 - 0.805i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.284 - 1.61i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.39 - 0.872i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (12.0 - 10.0i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (0.766 - 0.642i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.511 - 0.885i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.57 + 2.15i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (3.10 + 17.6i)T + (-91.1 + 33.1i)T^{2} \)
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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

−9.923244053080746414478100873401, −8.838265305755574653332321981468, −8.717170888002163242189990548690, −7.60889119799658380674827911498, −6.67293846734826652866442654912, −5.63910902698053525311282015810, −4.43914753725609726934529766603, −3.29377147504321938504406966702, −2.42694756267986886009669496741, −1.35470255630095508169848876718, 0.958441805206467784093265928495, 3.16023219447559000108401861847, 3.63935858797945197202978142971, 4.89122208013654344516639969902, 5.80218852173189709144647851890, 6.85437216608310591624168634530, 7.57182516450597417140768700375, 8.586297968425206071632138574059, 9.086437566295119517953047651736, 10.01646296531190529919465379232

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