↑

Properties

Label 2-61-61.52-c1-0-1
Degree $2$
Conductor $61$
Sign $0.0176 - 0.999i$
Analytic cond. $0.487087$
Root an. cond. $0.697916$
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  + (−1.09 + 1.50i)2-s + (2.38 + 1.73i)3-s + (−0.446 − 1.37i)4-s + (−1.04 − 3.21i)5-s + (−5.21 + 1.69i)6-s + (−0.380 − 0.524i)7-s + (−0.977 − 0.317i)8-s + (1.76 + 5.44i)9-s + (5.96 + 1.93i)10-s − 1.99i·11-s + (1.31 − 4.06i)12-s − 3.16·13-s + 1.20·14-s + (3.08 − 9.49i)15-s + (3.88 − 2.82i)16-s + (−1.97 + 0.641i)17-s + ⋯
L(s)  = 1  + (−0.771 + 1.06i)2-s + (1.37 + 1.00i)3-s + (−0.223 − 0.687i)4-s + (−0.467 − 1.43i)5-s + (−2.12 + 0.691i)6-s + (−0.144 − 0.198i)7-s + (−0.345 − 0.112i)8-s + (0.589 + 1.81i)9-s + (1.88 + 0.613i)10-s − 0.601i·11-s + (0.381 − 1.17i)12-s − 0.878·13-s + 0.321·14-s + (0.796 − 2.45i)15-s + (0.970 − 0.705i)16-s + (−0.478 + 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0176 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0176 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.0176 - 0.999i$
Analytic conductor: \(0.487087\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :1/2),\ 0.0176 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.580271 + 0.570108i\)
\(L(\frac12)\) \(\approx\) \(0.580271 + 0.570108i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (6.96 - 3.53i)T \)
good2 \( 1 + (1.09 - 1.50i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-2.38 - 1.73i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (1.04 + 3.21i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.380 + 0.524i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + 1.99iT - 11T^{2} \)
13 \( 1 + 3.16T + 13T^{2} \)
17 \( 1 + (1.97 - 0.641i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.52 - 4.74i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.69 - 0.551i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 - 4.71iT - 29T^{2} \)
31 \( 1 + (3.45 + 4.75i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.671 - 0.924i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.608 - 0.442i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (7.92 + 2.57i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + 0.0653T + 47T^{2} \)
53 \( 1 + (-8.91 - 2.89i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.23 + 4.45i)T + (-18.2 - 56.1i)T^{2} \)
67 \( 1 + (2.94 - 0.957i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.91 - 0.947i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.61 + 4.95i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.2 - 3.31i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.991 - 0.720i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.86 - 6.69i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (-6.74 + 4.89i)T + (29.9 - 92.2i)T^{2} \)
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

−15.56932535287215291580653853048, −14.67846167950596422507575590468, −13.51826532804822177081440269251, −12.09231940152374570502648891760, −9.967363795048455072956523650800, −9.156907244975055698230385633021, −8.375345325601308638728835063860, −7.57083980208252613396584152356, −5.23743660937168662190235466656, −3.65390006877344087148754636667, 2.32617001590758125321994768538, 3.18577171183283399854914983262, 6.87770491204540242513167629155, 7.70391455559488097608413290129, 9.101923422559737336654810582101, 10.03691160300117550786786319241, 11.42598119774973159874782329458, 12.32846995696405924930268203737, 13.67500500480185252846666184070, 14.71424571706603900761742997779

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