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Properties

Label 2-124-124.123-c3-0-26
Degree $2$
Conductor $124$
Sign $0.608 + 0.793i$
Analytic cond. $7.31623$
Root an. cond. $2.70485$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.546 − 2.77i)2-s + 7.92·3-s + (−7.40 + 3.03i)4-s + 15.6·5-s + (−4.32 − 21.9i)6-s + 9.09i·7-s + (12.4 + 18.8i)8-s + 35.7·9-s + (−8.56 − 43.5i)10-s + 3.40·11-s + (−58.6 + 24.0i)12-s + 11.0i·13-s + (25.2 − 4.96i)14-s + 124.·15-s + (45.6 − 44.8i)16-s + 59.9i·17-s + ⋯
L(s)  = 1  + (−0.193 − 0.981i)2-s + 1.52·3-s + (−0.925 + 0.378i)4-s + 1.40·5-s + (−0.294 − 1.49i)6-s + 0.491i·7-s + (0.550 + 0.834i)8-s + 1.32·9-s + (−0.270 − 1.37i)10-s + 0.0933·11-s + (−1.41 + 0.577i)12-s + 0.236i·13-s + (0.481 − 0.0948i)14-s + 2.13·15-s + (0.712 − 0.701i)16-s + 0.855i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(7.31623\)
Root analytic conductor: \(2.70485\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :3/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.34695 - 1.15794i\)
\(L(\frac12)\) \(\approx\) \(2.34695 - 1.15794i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.546 + 2.77i)T \)
31 \( 1 + (45.2 + 166. i)T \)
good3 \( 1 - 7.92T + 27T^{2} \)
5 \( 1 - 15.6T + 125T^{2} \)
7 \( 1 - 9.09iT - 343T^{2} \)
11 \( 1 - 3.40T + 1.33e3T^{2} \)
13 \( 1 - 11.0iT - 2.19e3T^{2} \)
17 \( 1 - 59.9iT - 4.91e3T^{2} \)
19 \( 1 + 97.2iT - 6.85e3T^{2} \)
23 \( 1 + 185.T + 1.21e4T^{2} \)
29 \( 1 + 216. iT - 2.43e4T^{2} \)
37 \( 1 - 44.9iT - 5.06e4T^{2} \)
41 \( 1 - 130.T + 6.89e4T^{2} \)
43 \( 1 + 396.T + 7.95e4T^{2} \)
47 \( 1 - 111. iT - 1.03e5T^{2} \)
53 \( 1 - 730. iT - 1.48e5T^{2} \)
59 \( 1 + 371. iT - 2.05e5T^{2} \)
61 \( 1 - 718. iT - 2.26e5T^{2} \)
67 \( 1 - 672. iT - 3.00e5T^{2} \)
71 \( 1 - 168. iT - 3.57e5T^{2} \)
73 \( 1 + 1.04e3iT - 3.89e5T^{2} \)
79 \( 1 - 661.T + 4.93e5T^{2} \)
83 \( 1 + 144.T + 5.71e5T^{2} \)
89 \( 1 + 734. iT - 7.04e5T^{2} \)
97 \( 1 - 343.T + 9.12e5T^{2} \)
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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

−13.14557966060764769479040821705, −11.82854394681881954699097065356, −10.32153008238701494227936806766, −9.532567770952551906197175446066, −8.899631916533946857972839085620, −7.88270034241672649193676558741, −5.93547712096095184332093673608, −4.15768291982936331684842148192, −2.61498556986448364926962684695, −1.89138035138570273009879469983, 1.78410680153999254819357823577, 3.61573890378022603144991084462, 5.31357298346605695078130774293, 6.63997377019784274367560240638, 7.80666262095796793408118061743, 8.746686330316660843326823614384, 9.704005873802380359898406041628, 10.25092362654692713443820871356, 12.66552553882473958391282147420, 13.71570654275992926597784052600

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