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Properties

Label 2-122-61.13-c1-0-0
Degree $2$
Conductor $122$
Sign $-0.760 - 0.649i$
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  + (−0.5 + 0.866i)2-s − 0.772·3-s + (−0.499 − 0.866i)4-s + (−1.20 + 2.08i)5-s + (0.386 − 0.669i)6-s + (−1.70 + 2.94i)7-s + 0.999·8-s − 2.40·9-s + (−1.20 − 2.08i)10-s + 4.17·11-s + (0.386 + 0.669i)12-s + (−1.92 + 3.34i)13-s + (−1.70 − 2.94i)14-s + (0.928 − 1.60i)15-s + (−0.5 + 0.866i)16-s + (0.314 + 0.545i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s − 0.446·3-s + (−0.249 − 0.433i)4-s + (−0.537 + 0.930i)5-s + (0.157 − 0.273i)6-s + (−0.643 + 1.11i)7-s + 0.353·8-s − 0.800·9-s + (−0.379 − 0.658i)10-s + 1.25·11-s + (0.111 + 0.193i)12-s + (−0.534 + 0.926i)13-s + (−0.454 − 0.787i)14-s + (0.239 − 0.415i)15-s + (−0.125 + 0.216i)16-s + (0.0763 + 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(122\)    =    \(2 \cdot 61\)
Sign: $-0.760 - 0.649i$
Analytic conductor: \(0.974174\)
Root analytic conductor: \(0.987002\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{122} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 122,\ (\ :1/2),\ -0.760 - 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192009 + 0.520421i\)
\(L(\frac12)\) \(\approx\) \(0.192009 + 0.520421i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (2.76 + 7.30i)T \)
good3 \( 1 + 0.772T + 3T^{2} \)
5 \( 1 + (1.20 - 2.08i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.70 - 2.94i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.17T + 11T^{2} \)
13 \( 1 + (1.92 - 3.34i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.314 - 0.545i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.78 + 6.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 9.43T + 23T^{2} \)
29 \( 1 + (-3.74 - 6.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.474 - 0.821i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.194T + 37T^{2} \)
41 \( 1 + 2.54T + 41T^{2} \)
43 \( 1 + (1.31 - 2.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.542 + 0.938i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + (0.542 - 0.938i)T + (-29.5 - 51.0i)T^{2} \)
67 \( 1 + (0.474 - 0.821i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.26 - 10.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.74 - 11.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.31 + 4.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.24 + 3.88i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.967T + 89T^{2} \)
97 \( 1 + (-4.22 - 7.31i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.28683412220510695666575856174, −12.68524163954773165558923077991, −11.57210242227790404374447397937, −10.94989627239518550637400783094, −9.281941830667367518977747161204, −8.745753555744528839210323406210, −6.83440868145110308478204880781, −6.54250825739338218861842312443, −4.96078925964005037477290585815, −2.97736057245018038464205723331, 0.72341688512513292509886879453, 3.46469712310274052657038968159, 4.71014031624772064758919908803, 6.39284542393003854836387662311, 7.84200387605045045123665353175, 8.871826167799055849448551235498, 10.02033227172388380701487756364, 10.99837240164668085571850558978, 12.08473441839062878083240502513, 12.67522629357916563383192861372

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