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Properties

Label 2-124-4.3-c2-0-8
Degree $2$
Conductor $124$
Sign $0.716 - 0.697i$
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  + (0.778 − 1.84i)2-s + 5.21i·3-s + (−2.78 − 2.86i)4-s + 3.92·5-s + (9.60 + 4.05i)6-s + 8.65i·7-s + (−7.45 + 2.90i)8-s − 18.2·9-s + (3.05 − 7.23i)10-s + 6.53i·11-s + (14.9 − 14.5i)12-s + 11.5·13-s + (15.9 + 6.73i)14-s + 20.4i·15-s + (−0.444 + 15.9i)16-s + 24.6·17-s + ⋯
L(s)  = 1  + (0.389 − 0.921i)2-s + 1.73i·3-s + (−0.697 − 0.716i)4-s + 0.785·5-s + (1.60 + 0.676i)6-s + 1.23i·7-s + (−0.931 + 0.363i)8-s − 2.02·9-s + (0.305 − 0.723i)10-s + 0.594i·11-s + (1.24 − 1.21i)12-s + 0.889·13-s + (1.13 + 0.480i)14-s + 1.36i·15-s + (−0.0277 + 0.999i)16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.716 - 0.697i$
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: \((2,\ 124,\ (\ :1),\ 0.716 - 0.697i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.55716 + 0.632360i\)
\(L(\frac12)\) \(\approx\) \(1.55716 + 0.632360i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.778 + 1.84i)T \)
31 \( 1 + 5.56iT \)
good3 \( 1 - 5.21iT - 9T^{2} \)
5 \( 1 - 3.92T + 25T^{2} \)
7 \( 1 - 8.65iT - 49T^{2} \)
11 \( 1 - 6.53iT - 121T^{2} \)
13 \( 1 - 11.5T + 169T^{2} \)
17 \( 1 - 24.6T + 289T^{2} \)
19 \( 1 + 25.8iT - 361T^{2} \)
23 \( 1 + 8.45iT - 529T^{2} \)
29 \( 1 + 30.8T + 841T^{2} \)
37 \( 1 - 14.6T + 1.36e3T^{2} \)
41 \( 1 - 41.0T + 1.68e3T^{2} \)
43 \( 1 + 84.0iT - 1.84e3T^{2} \)
47 \( 1 + 39.5iT - 2.20e3T^{2} \)
53 \( 1 + 47.9T + 2.80e3T^{2} \)
59 \( 1 - 55.6iT - 3.48e3T^{2} \)
61 \( 1 - 79.6T + 3.72e3T^{2} \)
67 \( 1 - 121. iT - 4.48e3T^{2} \)
71 \( 1 + 54.6iT - 5.04e3T^{2} \)
73 \( 1 - 35.9T + 5.32e3T^{2} \)
79 \( 1 + 18.7iT - 6.24e3T^{2} \)
83 \( 1 - 38.9iT - 6.88e3T^{2} \)
89 \( 1 + 66.4T + 7.92e3T^{2} \)
97 \( 1 - 174.T + 9.40e3T^{2} \)
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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

−13.28160848637092488554117772831, −12.03535041671158978353002040549, −11.10873529683106145161239435028, −10.13141749903602540202507900849, −9.406642038433414342246835976859, −8.748635483820876686208434293483, −5.80134173551576789845893312038, −5.22056449602350166594076544401, −3.84014668420006751088459694677, −2.49111157588672857848662049565, 1.21004392579137175029227374338, 3.51427059777651505477248345485, 5.74976554451422123246929982314, 6.31603126376585438513775154361, 7.60944912007426312566297581137, 8.053531508272845353731904317490, 9.661122776572039243582106126625, 11.28063367279400544434397079616, 12.60590007306823432622382357147, 13.19680074508755916277742449808

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