↑

Properties

Label 2-1950-65.49-c1-0-3
Degree $2$
Conductor $1950$
Sign $-0.577 - 0.816i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.499i)6-s + (−0.661 + 1.14i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−3.99 + 2.30i)11-s − 0.999i·12-s + (1.66 − 3.20i)13-s + 1.32·14-s + (−0.5 − 0.866i)16-s + (3.46 + 2i)17-s − 0.999·18-s + (−1.98 − 1.14i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.353 + 0.204i)6-s + (−0.249 + 0.432i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−1.20 + 0.695i)11-s − 0.288i·12-s + (0.460 − 0.887i)13-s + 0.353·14-s + (−0.125 − 0.216i)16-s + (0.840 + 0.485i)17-s − 0.235·18-s + (−0.455 − 0.262i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3739225468\)
\(L(\frac12)\) \(\approx\) \(0.3739225468\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-1.66 + 3.20i)T \)
good7 \( 1 + (0.661 - 1.14i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.99 - 2.30i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.98 + 1.14i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.50 + 4.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.01 + 1.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 + (-3.40 - 5.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.02 - 2.32i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.45 + 4.30i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.10T + 47T^{2} \)
53 \( 1 + 0.826iT - 53T^{2} \)
59 \( 1 + (2.72 + 1.57i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.267 - 0.464i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.59 + 2.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.81 + 5.66i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.28T + 73T^{2} \)
79 \( 1 + 2.96T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + (10.2 - 5.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.40 + 7.63i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.674604047634895094257591339004, −8.664394194955672706045876008261, −8.153635029331310719939125498586, −7.15167372123432376308186707414, −6.26809428502783370888027964482, −5.19880620579894216389256626823, −4.75615832446305458515610818094, −3.35684352014078973626308151218, −2.73856732174122696160938306624, −1.32110351723315694892945962564, 0.17791436845095923230113868851, 1.45899157530161177186398045257, 2.93999965875440290276380987972, 4.08347845625867929469574422137, 5.19591312225430681613062139106, 5.73531892447573890964180769902, 6.63351865641185619637318729618, 7.34566455347099416930804643679, 7.967189393578359726914607642724, 8.820231617553510705972665693590

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