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Properties

Label 2-124-124.99-c1-0-11
Degree $2$
Conductor $124$
Sign $0.884 + 0.466i$
Analytic cond. $0.990144$
Root an. cond. $0.995060$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.26 − 0.629i)2-s + (0.108 + 0.188i)3-s + (1.20 − 1.59i)4-s + (−0.924 + 1.60i)5-s + (0.256 + 0.170i)6-s + (0.987 − 0.569i)7-s + (0.528 − 2.77i)8-s + (1.47 − 2.55i)9-s + (−0.163 + 2.61i)10-s + (−3.13 + 5.42i)11-s + (0.432 + 0.0544i)12-s + (−3.05 − 1.76i)13-s + (0.891 − 1.34i)14-s − 0.402·15-s + (−1.07 − 3.85i)16-s + (−4.28 + 2.47i)17-s + ⋯
L(s)  = 1  + (0.895 − 0.444i)2-s + (0.0628 + 0.108i)3-s + (0.604 − 0.796i)4-s + (−0.413 + 0.716i)5-s + (0.104 + 0.0695i)6-s + (0.373 − 0.215i)7-s + (0.186 − 0.982i)8-s + (0.492 − 0.852i)9-s + (−0.0518 + 0.825i)10-s + (−0.944 + 1.63i)11-s + (0.124 + 0.0157i)12-s + (−0.848 − 0.489i)13-s + (0.238 − 0.358i)14-s − 0.103·15-s + (−0.269 − 0.962i)16-s + (−1.03 + 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1/2),\ 0.884 + 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57374 - 0.389888i\)
\(L(\frac12)\) \(\approx\) \(1.57374 - 0.389888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 + 0.629i)T \)
31 \( 1 + (4.57 + 3.16i)T \)
good3 \( 1 + (-0.108 - 0.188i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.924 - 1.60i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.987 + 0.569i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.13 - 5.42i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.05 + 1.76i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.28 - 2.47i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.94 - 1.12i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.07T + 23T^{2} \)
29 \( 1 + 0.693iT - 29T^{2} \)
37 \( 1 + (-5.84 + 3.37i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.73 + 3.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.612 + 1.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.29iT - 47T^{2} \)
53 \( 1 + (-0.805 - 0.464i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.2 + 5.93i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 3.76iT - 61T^{2} \)
67 \( 1 + (-2.14 - 1.24i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.08 - 4.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.63 - 3.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.96 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.17 - 7.22i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.6iT - 89T^{2} \)
97 \( 1 - 9.12T + 97T^{2} \)
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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

−12.91559877393738129442364156668, −12.65386108706479953412034614491, −11.26651227118968910519521613818, −10.47805932341992057068325081522, −9.527036089728520766757292875345, −7.48987669027942398779648311409, −6.77667581266615161253554334438, −5.04528085492748747655106037876, −3.96616256731174504193980952611, −2.38438154684881926097413446099, 2.67019438920017861105639767236, 4.57178375684924386587973043670, 5.27410455083919806713108107867, 6.89651425474859604696440454208, 8.027580580253490082283020482403, 8.820006454943688136217497860002, 10.84986431697977771661617341474, 11.52782504856787280034274653077, 12.87842610763067241067558364449, 13.31410165281144620882429881108

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