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Properties

Label 2-124-31.25-c3-0-5
Degree $2$
Conductor $124$
Sign $-0.222 + 0.974i$
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  + (0.269 + 0.466i)3-s + (−7.15 + 12.3i)5-s + (−17.6 − 30.5i)7-s + (13.3 − 23.1i)9-s + (−5.05 + 8.74i)11-s + (42.3 − 73.4i)13-s − 7.71·15-s + (13.9 + 24.1i)17-s + (−34.6 − 60.0i)19-s + (9.51 − 16.4i)21-s − 162.·23-s + (−39.9 − 69.2i)25-s + 28.9·27-s − 134.·29-s + (101. − 139. i)31-s + ⋯
L(s)  = 1  + (0.0518 + 0.0898i)3-s + (−0.640 + 1.10i)5-s + (−0.952 − 1.65i)7-s + (0.494 − 0.856i)9-s + (−0.138 + 0.239i)11-s + (0.904 − 1.56i)13-s − 0.132·15-s + (0.198 + 0.344i)17-s + (−0.418 − 0.725i)19-s + (0.0988 − 0.171i)21-s − 1.47·23-s + (−0.319 − 0.553i)25-s + 0.206·27-s − 0.861·29-s + (0.585 − 0.810i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 + 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.222 + 0.974i$
Analytic conductor: \(7.31623\)
Root analytic conductor: \(2.70485\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :3/2),\ -0.222 + 0.974i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.565872 - 0.709786i\)
\(L(\frac12)\) \(\approx\) \(0.565872 - 0.709786i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 + (-101. + 139. i)T \)
good3 \( 1 + (-0.269 - 0.466i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (7.15 - 12.3i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (17.6 + 30.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (5.05 - 8.74i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-42.3 + 73.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-13.9 - 24.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (34.6 + 60.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 162.T + 1.21e4T^{2} \)
29 \( 1 + 134.T + 2.43e4T^{2} \)
37 \( 1 + (-18.5 - 32.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (154. - 267. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (126. + 218. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 277.T + 1.03e5T^{2} \)
53 \( 1 + (-215. + 373. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-271. - 470. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 - 805.T + 2.26e5T^{2} \)
67 \( 1 + (36.6 - 63.4i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-371. + 643. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (126. - 219. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-179. - 310. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-631. + 1.09e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 386.T + 7.04e5T^{2} \)
97 \( 1 - 446.T + 9.12e5T^{2} \)
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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

−12.87483322313712981535111614477, −11.41783807948707786964115230243, −10.36292732219639082752990057178, −9.957984919860305673711642264237, −8.057619573126679225774821947342, −7.08900024772267991236948088575, −6.26163103954437808559784985925, −3.95157916287794395290831023090, −3.33641940741117546267600430054, −0.46766880413749423594827292789, 1.96700783369464821996074467744, 3.91822310402080053081146046794, 5.30109147693904902881326298805, 6.48820860954961457356536304624, 8.184850205090680825426917740787, 8.830054177715683718621770622101, 9.863235131328860589116523803645, 11.51615728671754564910360996624, 12.24448233471649438479179092020, 13.00135681140240104731094413402

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