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Properties

Label 2-124-124.123-c3-0-5
Degree $2$
Conductor $124$
Sign $0.513 - 0.857i$
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  + (−1.77 + 2.20i)2-s − 7.49·3-s + (−1.71 − 7.81i)4-s − 18.0·5-s + (13.2 − 16.5i)6-s − 21.7i·7-s + (20.2 + 10.0i)8-s + 29.2·9-s + (32.0 − 39.8i)10-s − 59.2·11-s + (12.8 + 58.5i)12-s + 62.3i·13-s + (47.9 + 38.5i)14-s + 135.·15-s + (−58.0 + 26.8i)16-s − 0.447i·17-s + ⋯
L(s)  = 1  + (−0.626 + 0.779i)2-s − 1.44·3-s + (−0.214 − 0.976i)4-s − 1.61·5-s + (0.904 − 1.12i)6-s − 1.17i·7-s + (0.895 + 0.444i)8-s + 1.08·9-s + (1.01 − 1.26i)10-s − 1.62·11-s + (0.310 + 1.40i)12-s + 1.32i·13-s + (0.915 + 0.736i)14-s + 2.33·15-s + (−0.907 + 0.419i)16-s − 0.00637i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.513 - 0.857i$
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: \((2,\ 124,\ (\ :3/2),\ 0.513 - 0.857i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.185933 + 0.105394i\)
\(L(\frac12)\) \(\approx\) \(0.185933 + 0.105394i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.77 - 2.20i)T \)
31 \( 1 + (-125. - 118. i)T \)
good3 \( 1 + 7.49T + 27T^{2} \)
5 \( 1 + 18.0T + 125T^{2} \)
7 \( 1 + 21.7iT - 343T^{2} \)
11 \( 1 + 59.2T + 1.33e3T^{2} \)
13 \( 1 - 62.3iT - 2.19e3T^{2} \)
17 \( 1 + 0.447iT - 4.91e3T^{2} \)
19 \( 1 + 93.8iT - 6.85e3T^{2} \)
23 \( 1 + 46.9T + 1.21e4T^{2} \)
29 \( 1 - 207. iT - 2.43e4T^{2} \)
37 \( 1 + 353. iT - 5.06e4T^{2} \)
41 \( 1 + 59.8T + 6.89e4T^{2} \)
43 \( 1 - 5.12T + 7.95e4T^{2} \)
47 \( 1 + 321. iT - 1.03e5T^{2} \)
53 \( 1 + 494. iT - 1.48e5T^{2} \)
59 \( 1 - 225. iT - 2.05e5T^{2} \)
61 \( 1 - 177. iT - 2.26e5T^{2} \)
67 \( 1 - 50.0iT - 3.00e5T^{2} \)
71 \( 1 - 451. iT - 3.57e5T^{2} \)
73 \( 1 - 687. iT - 3.89e5T^{2} \)
79 \( 1 + 1.03e3T + 4.93e5T^{2} \)
83 \( 1 - 75.5T + 5.71e5T^{2} \)
89 \( 1 + 95.8iT - 7.04e5T^{2} \)
97 \( 1 - 389.T + 9.12e5T^{2} \)
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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

−13.01519717543028506666089018441, −11.68218917314665115148567146028, −10.95310067165868918484887576831, −10.31386879393152647213029646540, −8.559170540004919198141009033684, −7.31209389012376722648598783511, −6.89536471704268636275626756775, −5.18543810358883933268235713134, −4.29912320277493890883014085924, −0.55189448072889186784935436770, 0.35697387050869210690828951704, 2.95149515570368708961542538427, 4.64235304641375944113291810757, 5.87330579032997494136015102053, 7.81014126657603295085346312412, 8.178766940110868838091598002835, 10.07848574578278876159888459339, 10.84893593182958327186912596913, 11.80906010467456906671993560211, 12.20020640461253714709673346381

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