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Properties

Label 2-122-61.14-c1-0-1
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + 1.25·3-s + (0.499 − 0.866i)4-s + (−1.99 − 3.44i)5-s + (−1.08 + 0.625i)6-s + (4.15 − 2.39i)7-s + 0.999i·8-s − 1.43·9-s + (3.44 + 1.99i)10-s + 5.15i·11-s + (0.625 − 1.08i)12-s + (−0.125 − 0.217i)13-s + (−2.39 + 4.15i)14-s + (−2.49 − 4.31i)15-s + (−0.5 − 0.866i)16-s + (3.55 + 2.04i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + 0.722·3-s + (0.249 − 0.433i)4-s + (−0.890 − 1.54i)5-s + (−0.442 + 0.255i)6-s + (1.57 − 0.906i)7-s + 0.353i·8-s − 0.478·9-s + (1.09 + 0.629i)10-s + 1.55i·11-s + (0.180 − 0.312i)12-s + (−0.0348 − 0.0603i)13-s + (−0.641 + 1.11i)14-s + (−0.643 − 1.11i)15-s + (−0.125 − 0.216i)16-s + (0.861 + 0.497i)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

\(L(1)\) \(\approx\) \(0.926083 - 0.237575i\)
\(L(\frac12)\) \(\approx\) \(0.926083 - 0.237575i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.25T + 3T^{2} \)
5 \( 1 + (1.99 + 3.44i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-4.15 + 2.39i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 5.15iT - 11T^{2} \)
13 \( 1 + (0.125 + 0.217i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.55 - 2.04i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.441 + 0.765i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.451iT - 23T^{2} \)
29 \( 1 + (2.07 + 1.19i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.575 - 0.332i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.33iT - 37T^{2} \)
41 \( 1 - 3.96T + 41T^{2} \)
43 \( 1 + (0.590 - 0.340i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.40 - 2.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.83iT - 53T^{2} \)
59 \( 1 + (-6.30 + 3.64i)T + (29.5 - 51.0i)T^{2} \)
67 \( 1 + (8.67 - 5.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.35 + 3.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.447 + 0.774i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.28 + 3.05i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.89 - 5.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.71iT - 89T^{2} \)
97 \( 1 + (6.59 - 11.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.48069971907989301500574513406, −12.22101311752152678442353918951, −11.36682123529597257184281988980, −9.956481702022245237041053805945, −8.774100999047371835427586569868, −7.987740378371533626579062290465, −7.51141616043132813890906495692, −5.12940269115740282567416484414, −4.23129119662591843600499065446, −1.48355852495103935588930842077, 2.53527550436081937417560943188, 3.53152641125695464751579056555, 5.77303412522377434497879768446, 7.55492982546181724732578998858, 8.141087336829369061992978761732, 9.045769023292205039825122425712, 10.74651820084487681530125427055, 11.33181117603654904145392046678, 11.94073387967604224708602103890, 13.98986216896043774151618849765

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