Properties

Label 2-124-31.25-c3-0-4
Degree $2$
Conductor $124$
Sign $0.415 + 0.909i$
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  + (−1.36 − 2.37i)3-s + (−0.238 + 0.413i)5-s + (7.63 + 13.2i)7-s + (9.74 − 16.8i)9-s + (24.8 − 42.9i)11-s + (4.05 − 7.02i)13-s + 1.30·15-s + (−25.1 − 43.5i)17-s + (−81.6 − 141. i)19-s + (20.8 − 36.1i)21-s + 200.·23-s + (62.3 + 108. i)25-s − 127.·27-s + 90.6·29-s + (−4.76 − 172. i)31-s + ⋯
L(s)  = 1  + (−0.263 − 0.456i)3-s + (−0.0213 + 0.0369i)5-s + (0.412 + 0.713i)7-s + (0.361 − 0.625i)9-s + (0.680 − 1.17i)11-s + (0.0865 − 0.149i)13-s + 0.0225·15-s + (−0.358 − 0.621i)17-s + (−0.986 − 1.70i)19-s + (0.217 − 0.376i)21-s + 1.81·23-s + (0.499 + 0.864i)25-s − 0.907·27-s + 0.580·29-s + (−0.0275 − 0.999i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.415 + 0.909i$
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: \((2,\ 124,\ (\ :3/2),\ 0.415 + 0.909i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.28575 - 0.826016i\)
\(L(\frac12)\) \(\approx\) \(1.28575 - 0.826016i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.76 + 172. i)T \)
good3 \( 1 + (1.36 + 2.37i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (0.238 - 0.413i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-7.63 - 13.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-24.8 + 42.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-4.05 + 7.02i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (25.1 + 43.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (81.6 + 141. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 200.T + 1.21e4T^{2} \)
29 \( 1 - 90.6T + 2.43e4T^{2} \)
37 \( 1 + (-105. - 183. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (131. - 228. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-42.2 - 73.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 19.3T + 1.03e5T^{2} \)
53 \( 1 + (229. - 396. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-34.1 - 59.2i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + 573.T + 2.26e5T^{2} \)
67 \( 1 + (-446. + 773. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (487. - 844. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (475. - 823. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-149. - 259. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-133. + 231. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 901.T + 7.04e5T^{2} \)
97 \( 1 - 29.7T + 9.12e5T^{2} \)
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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

−12.81531914850111810908125777723, −11.52482298479584481354953886558, −11.12442765276701762483695769692, −9.289945081842506073367355222439, −8.649648250631638189860235307417, −7.06206274965311867817405005406, −6.17660683049678681764777176319, −4.75736660872937620733123680479, −2.93658812481736647214070432391, −0.924593300475784502193222957446, 1.69396607445187412624788288372, 4.03173033948351419819345584826, 4.86264492806207511913739963500, 6.55234096605932899827467920393, 7.65600227994254575911285542721, 8.933067986189251472293408145011, 10.33567175814568510577943494628, 10.70742053636373136266037865940, 12.14170688163995281669782149413, 13.01056948701272088874061781338

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