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Properties

Label 2-950-95.2-c1-0-19
Degree $2$
Conductor $950$
Sign $-0.0533 + 0.998i$
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  + (−0.573 − 0.819i)2-s + (−0.133 − 1.52i)3-s + (−0.342 + 0.939i)4-s + (−1.17 + 0.984i)6-s + (0.227 − 0.848i)7-s + (0.965 − 0.258i)8-s + (0.642 − 0.113i)9-s + (−1.39 + 2.40i)11-s + (1.47 + 0.396i)12-s + (5.85 + 0.512i)13-s + (−0.825 + 0.300i)14-s + (−0.766 − 0.642i)16-s + (0.437 − 0.306i)17-s + (−0.461 − 0.461i)18-s + (3.66 + 2.35i)19-s + ⋯
L(s)  = 1  + (−0.405 − 0.579i)2-s + (−0.0770 − 0.881i)3-s + (−0.171 + 0.469i)4-s + (−0.479 + 0.402i)6-s + (0.0859 − 0.320i)7-s + (0.341 − 0.0915i)8-s + (0.214 − 0.0377i)9-s + (−0.419 + 0.726i)11-s + (0.427 + 0.114i)12-s + (1.62 + 0.142i)13-s + (−0.220 + 0.0802i)14-s + (−0.191 − 0.160i)16-s + (0.106 − 0.0742i)17-s + (−0.108 − 0.108i)18-s + (0.841 + 0.539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0533 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0533 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.0533 + 0.998i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.0533 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953793 - 1.00607i\)
\(L(\frac12)\) \(\approx\) \(0.953793 - 1.00607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.573 + 0.819i)T \)
5 \( 1 \)
19 \( 1 + (-3.66 - 2.35i)T \)
good3 \( 1 + (0.133 + 1.52i)T + (-2.95 + 0.520i)T^{2} \)
7 \( 1 + (-0.227 + 0.848i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.39 - 2.40i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.85 - 0.512i)T + (12.8 + 2.25i)T^{2} \)
17 \( 1 + (-0.437 + 0.306i)T + (5.81 - 15.9i)T^{2} \)
23 \( 1 + (-5.86 + 2.73i)T + (14.7 - 17.6i)T^{2} \)
29 \( 1 + (0.259 + 1.47i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (3.95 - 2.28i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.22 - 4.22i)T - 37iT^{2} \)
41 \( 1 + (-2.19 + 2.61i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.10 + 6.64i)T + (-27.6 - 32.9i)T^{2} \)
47 \( 1 + (1.04 - 1.49i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (1.40 + 3.01i)T + (-34.0 + 40.6i)T^{2} \)
59 \( 1 + (-1.12 + 6.39i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (0.643 + 0.234i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.786 + 0.550i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (1.03 + 2.83i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (-9.60 + 0.840i)T + (71.8 - 12.6i)T^{2} \)
79 \( 1 + (1.70 + 1.42i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (12.7 + 3.41i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (10.4 - 8.78i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (3.87 + 5.53i)T + (-33.1 + 91.1i)T^{2} \)
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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

−9.917718615346919355928865759155, −8.973244869172580444893816556115, −8.149366324319753463165910529237, −7.31849603595036066483711581863, −6.71292103134425380405276155372, −5.53140796996943310083112160184, −4.28860755921869279463610103696, −3.26778206334299625391888448627, −1.88328434694967751498170273560, −0.980553564630072993206964842215, 1.20533214749480138460954730762, 3.10911866372711554529751709959, 4.07238909961126799783492693681, 5.27357230129774546338173304865, 5.75082626085644623888202603775, 6.92176387036434122899113444265, 7.81043163106427375432623214307, 8.820342290991214943222835922769, 9.207064403989339667044751615044, 10.18984143517111035840049351285

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