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Properties

Label 2-1950-13.9-c1-0-23
Degree $2$
Conductor $1950$
Sign $0.859 - 0.511i$
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 + (1.5 + 2.59i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 − 2.59i)11-s + 0.999·12-s + (2.5 − 2.59i)13-s − 3·14-s + (−0.5 + 0.866i)16-s + 0.999·18-s + (−1.5 − 2.59i)19-s − 3·21-s + (1.5 + 2.59i)22-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 + (0.566 + 0.981i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.452 − 0.783i)11-s + 0.288·12-s + (0.693 − 0.720i)13-s − 0.801·14-s + (−0.125 + 0.216i)16-s + 0.235·18-s + (−0.344 − 0.596i)19-s − 0.654·21-s + (0.319 + 0.553i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.366030010\)
\(L(\frac12)\) \(\approx\) \(1.366030010\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 2.59i)T \)
good7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + (-4.5 + 7.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5 + 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7 - 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4 - 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.060429347822173822436809890929, −8.527084788956893696958218148730, −7.914192362404614805719572018465, −6.79264625458891936640621092117, −5.81957153357429168895677594575, −5.60031059973353453200619198592, −4.50752833785881212612706935171, −3.53580232822593334805412243621, −2.25969933927803109755320388676, −0.72436030524830028311236538614, 1.09027320388650284101249275930, 1.77056853878608768810911377284, 3.08653212505917891754161351994, 4.38203075637276440088742411357, 4.60824271378925341149215040773, 6.27381733532605385172864484989, 6.69120006159615784795811198113, 7.82623802703187335432621076859, 8.145359499102354543690118284547, 9.220679743937605790105560463887

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