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Properties

Label 2-124-31.5-c3-0-4
Degree $2$
Conductor $124$
Sign $0.677 + 0.735i$
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  + (−4.95 + 8.58i)3-s + (−7.40 − 12.8i)5-s + (4.04 − 7.00i)7-s + (−35.5 − 61.6i)9-s + (18.9 + 32.7i)11-s + (13.5 + 23.4i)13-s + 146.·15-s + (39.1 − 67.8i)17-s + (72.0 − 124. i)19-s + (40.0 + 69.4i)21-s − 114.·23-s + (−47.2 + 81.9i)25-s + 437.·27-s − 86.1·29-s + (−157. − 71.1i)31-s + ⋯
L(s)  = 1  + (−0.953 + 1.65i)3-s + (−0.662 − 1.14i)5-s + (0.218 − 0.378i)7-s + (−1.31 − 2.28i)9-s + (0.518 + 0.897i)11-s + (0.289 + 0.500i)13-s + 2.52·15-s + (0.559 − 0.968i)17-s + (0.870 − 1.50i)19-s + (0.416 + 0.721i)21-s − 1.03·23-s + (−0.378 + 0.655i)25-s + 3.12·27-s − 0.551·29-s + (−0.911 − 0.412i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.677 + 0.735i$
Analytic conductor: \(7.31623\)
Root analytic conductor: \(2.70485\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :3/2),\ 0.677 + 0.735i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.680753 - 0.298275i\)
\(L(\frac12)\) \(\approx\) \(0.680753 - 0.298275i\)
\(L(\frac{5}{2})\) 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 \( 1 + (157. + 71.1i)T \)
good3 \( 1 + (4.95 - 8.58i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (7.40 + 12.8i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-4.04 + 7.00i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-18.9 - 32.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-13.5 - 23.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-39.1 + 67.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-72.0 + 124. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 114.T + 1.21e4T^{2} \)
29 \( 1 + 86.1T + 2.43e4T^{2} \)
37 \( 1 + (-170. + 295. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (157. + 273. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-163. + 282. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 58.1T + 1.03e5T^{2} \)
53 \( 1 + (133. + 231. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (279. - 483. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + 234.T + 2.26e5T^{2} \)
67 \( 1 + (107. + 186. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (399. + 692. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-193. - 335. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (73.5 - 127. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-429. - 744. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 284.T + 7.04e5T^{2} \)
97 \( 1 + 59.2T + 9.12e5T^{2} \)
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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

−12.30396182791771010849021335665, −11.70507864869767159132420849029, −10.82471279939999474916303405537, −9.482280697501062132812546250104, −9.091136425890592175145760399836, −7.30488353668289636726951359551, −5.54755201970911143620286186863, −4.61109618319801313502863907468, −3.90876480491464682567992933877, −0.47625497424652110167381119921, 1.44688396129565452380119619061, 3.30009355621881276793262722632, 5.75797502988770780629640902668, 6.32707891594399470706039058773, 7.64077959190845141654524492465, 8.156525879124409766618394149989, 10.40708354474997364187032125007, 11.37152212130216111631547363158, 11.89274419603198056570858196264, 12.85570838285241817197437095206

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