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Properties

Label 2-1950-13.3-c1-0-23
Degree $2$
Conductor $1950$
Sign $-0.202 - 0.979i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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 + (1.36 − 2.36i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.366 − 0.633i)11-s − 0.999·12-s + (−0.232 + 3.59i)13-s + 2.73·14-s + (−0.5 − 0.866i)16-s + (2.23 − 3.86i)17-s − 0.999·18-s + (−0.633 + 1.09i)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 + (0.516 − 0.894i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.110 − 0.191i)11-s − 0.288·12-s + (−0.0643 + 0.997i)13-s + 0.730·14-s + (−0.125 − 0.216i)16-s + (0.541 − 0.937i)17-s − 0.235·18-s + (−0.145 + 0.251i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.202 - 0.979i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.346057597\)
\(L(\frac12)\) \(\approx\) \(2.346057597\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (0.232 - 3.59i)T \)
good7 \( 1 + (-1.36 + 2.36i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.366 + 0.633i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.23 + 3.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.633 - 1.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.09 - 5.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.23 - 5.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3.96 - 6.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.59 - 7.96i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.63 - 2.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.66T + 47T^{2} \)
53 \( 1 + 7.73T + 53T^{2} \)
59 \( 1 + (-6.19 + 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.06 + 8.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.83 - 6.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.633 - 1.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 4.66T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 5.26T + 83T^{2} \)
89 \( 1 + (-2.46 - 4.26i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.570359221571557059542810648105, −8.393620296080588019688529311825, −7.88357785342434779271989359988, −7.03058868286751895464099871809, −6.36203185160931943859186275328, −5.06470395111084144661073976653, −4.72592463605759589454873799262, −3.72471667165049509052633006044, −2.89446939770363416851386006523, −1.28911704080273454261365840138, 0.820771662027204888197696014781, 2.16650057307795582869863523196, 2.74422042506076475094974567877, 3.87976489547270539604290728719, 4.90769610475150645729936164571, 5.70499125722044162871030313679, 6.40142870204875971390998903245, 7.56162434245775259836272953696, 8.324540576681515212155035348229, 8.820424217333315020464966976341

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