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Properties

Label 2-124-31.5-c3-0-3
Degree $2$
Conductor $124$
Sign $0.695 - 0.718i$
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  + (2.5 − 4.33i)3-s + (7.5 + 12.9i)5-s + (−14.5 + 25.1i)7-s + (0.999 + 1.73i)9-s + (25.5 + 44.1i)11-s + (−26.5 − 45.8i)13-s + 75·15-s + (−25.5 + 44.1i)17-s + (72.5 − 125. i)19-s + (72.5 + 125. i)21-s + 108·23-s + (−50 + 86.6i)25-s + 144.·27-s − 198·29-s + (−62 + 161. i)31-s + ⋯
L(s)  = 1  + (0.481 − 0.833i)3-s + (0.670 + 1.16i)5-s + (−0.782 + 1.35i)7-s + (0.0370 + 0.0641i)9-s + (0.698 + 1.21i)11-s + (−0.565 − 0.979i)13-s + 1.29·15-s + (−0.363 + 0.630i)17-s + (0.875 − 1.51i)19-s + (0.753 + 1.30i)21-s + 0.979·23-s + (−0.400 + 0.692i)25-s + 1.03·27-s − 1.26·29-s + (−0.359 + 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.695 - 0.718i$
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: \((2,\ 124,\ (\ :3/2),\ 0.695 - 0.718i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.76711 + 0.749466i\)
\(L(\frac12)\) \(\approx\) \(1.76711 + 0.749466i\)
\(L(\frac{5}{2})\) 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 \( 1 + (62 - 161. i)T \)
good3 \( 1 + (-2.5 + 4.33i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-7.5 - 12.9i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (14.5 - 25.1i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-25.5 - 44.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (26.5 + 45.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (25.5 - 44.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-72.5 + 125. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 108T + 1.21e4T^{2} \)
29 \( 1 + 198T + 2.43e4T^{2} \)
37 \( 1 + (-27.5 + 47.6i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (19.5 + 33.7i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-108.5 + 187. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 504T + 1.03e5T^{2} \)
53 \( 1 + (118.5 + 205. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-298.5 + 517. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + 466T + 2.26e5T^{2} \)
67 \( 1 + (374.5 + 648. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-160.5 - 277. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-396.5 - 686. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (608.5 - 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-277.5 - 480. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 378T + 7.04e5T^{2} \)
97 \( 1 - 1.07e3T + 9.12e5T^{2} \)
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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

−12.91925158582011081544174509236, −12.42509503778786845359081924434, −10.97287805649269812241432841812, −9.767280083210063945462366233794, −8.946467009436853455538719705570, −7.29284962641833792884670510295, −6.72347216269254547419069962059, −5.37361308200593026674738909726, −2.94564943221072461232430167602, −2.13232973208578404616792345410, 1.02691126080557645004569121374, 3.51017479665169200658415147616, 4.44695695231254433285938555412, 5.96745485608797165473502764129, 7.37952036965690991038176824485, 9.167954829501575427924128677101, 9.307572713288341011608608350999, 10.42007009906316648039681392183, 11.78178699995287012246224750745, 13.07801116080491758500047915576

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