Properties

Label 2-124-4.3-c2-0-27
Degree $2$
Conductor $124$
Sign $-0.998 + 0.0540i$
Analytic cond. $3.37875$
Root an. cond. $1.83813$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.37 + 1.45i)2-s − 5.29i·3-s + (−0.216 − 3.99i)4-s − 4.75·5-s + (7.69 + 7.28i)6-s − 1.34i·7-s + (6.09 + 5.18i)8-s − 19.0·9-s + (6.53 − 6.90i)10-s + 16.3i·11-s + (−21.1 + 1.14i)12-s − 13.1·13-s + (1.95 + 1.85i)14-s + 25.1i·15-s + (−15.9 + 1.72i)16-s + 5.86·17-s + ⋯
L(s)  = 1  + (−0.687 + 0.725i)2-s − 1.76i·3-s + (−0.0540 − 0.998i)4-s − 0.950·5-s + (1.28 + 1.21i)6-s − 0.192i·7-s + (0.762 + 0.647i)8-s − 2.11·9-s + (0.653 − 0.690i)10-s + 1.48i·11-s + (−1.76 + 0.0954i)12-s − 1.01·13-s + (0.139 + 0.132i)14-s + 1.67i·15-s + (−0.994 + 0.107i)16-s + 0.345·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0540i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.998 + 0.0540i$
Analytic conductor: \(3.37875\)
Root analytic conductor: \(1.83813\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1),\ -0.998 + 0.0540i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00812626 - 0.300647i\)
\(L(\frac12)\) \(\approx\) \(0.00812626 - 0.300647i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 1.45i)T \)
31 \( 1 - 5.56iT \)
good3 \( 1 + 5.29iT - 9T^{2} \)
5 \( 1 + 4.75T + 25T^{2} \)
7 \( 1 + 1.34iT - 49T^{2} \)
11 \( 1 - 16.3iT - 121T^{2} \)
13 \( 1 + 13.1T + 169T^{2} \)
17 \( 1 - 5.86T + 289T^{2} \)
19 \( 1 + 17.9iT - 361T^{2} \)
23 \( 1 + 33.6iT - 529T^{2} \)
29 \( 1 + 15.6T + 841T^{2} \)
37 \( 1 + 35.3T + 1.36e3T^{2} \)
41 \( 1 + 32.7T + 1.68e3T^{2} \)
43 \( 1 + 70.3iT - 1.84e3T^{2} \)
47 \( 1 - 68.9iT - 2.20e3T^{2} \)
53 \( 1 - 19.8T + 2.80e3T^{2} \)
59 \( 1 + 57.3iT - 3.48e3T^{2} \)
61 \( 1 + 115.T + 3.72e3T^{2} \)
67 \( 1 + 85.3iT - 4.48e3T^{2} \)
71 \( 1 + 23.9iT - 5.04e3T^{2} \)
73 \( 1 - 101.T + 5.32e3T^{2} \)
79 \( 1 + 65.4iT - 6.24e3T^{2} \)
83 \( 1 + 10.8iT - 6.88e3T^{2} \)
89 \( 1 - 66.9T + 7.92e3T^{2} \)
97 \( 1 - 168.T + 9.40e3T^{2} \)
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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

−12.50408659455144062604900198891, −11.95131359150077201665386909749, −10.56969607343525610440931353941, −9.087667791135409758609470625054, −7.85191820007209836163866525356, −7.32247722677633495553562304884, −6.60008339457408828329561361682, −4.88856733207450819699856171581, −2.10695384156154525492913293290, −0.25114535690379505774372507245, 3.22447907635202585478736516848, 3.95305934313305976614365218729, 5.43539436342691533780488393624, 7.74183078368691815129333313766, 8.681841445947284814300376666987, 9.639836436425985982715170019874, 10.49528307984663912993702276920, 11.44246753646045875816701091098, 11.99171786375205095887825771956, 13.71891618080587683097003905047

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