↑

Properties

Label 2-122-61.14-c1-0-3
Degree $2$
Conductor $122$
Sign $0.876 + 0.481i$
Analytic cond. $0.974174$
Root an. cond. $0.987002$
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  + (0.866 − 0.5i)2-s + 1.87·3-s + (0.499 − 0.866i)4-s + (−0.571 − 0.989i)5-s + (1.62 − 0.937i)6-s + (−3.92 + 2.26i)7-s − 0.999i·8-s + 0.515·9-s + (−0.989 − 0.571i)10-s + 3.10i·11-s + (0.937 − 1.62i)12-s + (−0.437 − 0.757i)13-s + (−2.26 + 3.92i)14-s + (−1.07 − 1.85i)15-s + (−0.5 − 0.866i)16-s + (3.90 + 2.25i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + 1.08·3-s + (0.249 − 0.433i)4-s + (−0.255 − 0.442i)5-s + (0.662 − 0.382i)6-s + (−1.48 + 0.856i)7-s − 0.353i·8-s + 0.171·9-s + (−0.312 − 0.180i)10-s + 0.936i·11-s + (0.270 − 0.468i)12-s + (−0.121 − 0.210i)13-s + (−0.605 + 1.04i)14-s + (−0.276 − 0.479i)15-s + (−0.125 − 0.216i)16-s + (0.947 + 0.547i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(122\)    =    \(2 \cdot 61\)
Sign: $0.876 + 0.481i$
Analytic conductor: \(0.974174\)
Root analytic conductor: \(0.987002\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{122} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 122,\ (\ :1/2),\ 0.876 + 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59941 - 0.410311i\)
\(L(\frac12)\) \(\approx\) \(1.59941 - 0.410311i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-3.46 + 7i)T \)
good3 \( 1 - 1.87T + 3T^{2} \)
5 \( 1 + (0.571 + 0.989i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.92 - 2.26i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 3.10iT - 11T^{2} \)
13 \( 1 + (0.437 + 0.757i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.90 - 2.25i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.32 + 5.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.34iT - 23T^{2} \)
29 \( 1 + (6.69 + 3.86i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.19 - 2.99i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.49iT - 37T^{2} \)
41 \( 1 + 4.21T + 41T^{2} \)
43 \( 1 + (2.71 - 1.57i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.69 - 4.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.24iT - 53T^{2} \)
59 \( 1 + (4.69 - 2.71i)T + (29.5 - 51.0i)T^{2} \)
67 \( 1 + (-13.2 + 7.67i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.2 - 6.46i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.23 - 9.07i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.90 - 3.40i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.66 - 11.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.41iT - 89T^{2} \)
97 \( 1 + (-2.01 + 3.48i)T + (-48.5 - 84.0i)T^{2} \)
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

−13.18360063354722191748521311663, −12.61602448081258170482833409520, −11.66993009603998282032663639048, −9.881281693716087047024118414459, −9.352738959775938407691342014199, −8.112399776018138001415405422868, −6.66992333165165640018984505161, −5.23977354266893630590735840005, −3.56694633331269308182771057091, −2.56142078309213642755612623017, 3.18788653721355414085647057433, 3.59459084553609210248220391588, 5.77890291496461229465065465650, 7.04512974046625132590129559923, 7.915675479882686888968978643976, 9.253841161874912189952356637341, 10.26849563728209428937073490442, 11.65461806322140130699217089988, 12.89930335890385583460734970917, 13.77203628796082127427919120669

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