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Properties

Label 2-124-4.3-c2-0-25
Degree $2$
Conductor $124$
Sign $0.115 + 0.993i$
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  + (1.99 − 0.115i)2-s − 2.95i·3-s + (3.97 − 0.462i)4-s − 7.62·5-s + (−0.342 − 5.90i)6-s − 12.8i·7-s + (7.87 − 1.38i)8-s + 0.249·9-s + (−15.2 + 0.882i)10-s + 15.4i·11-s + (−1.36 − 11.7i)12-s + 18.2·13-s + (−1.48 − 25.6i)14-s + 22.5i·15-s + (15.5 − 3.67i)16-s − 8.21·17-s + ⋯
L(s)  = 1  + (0.998 − 0.0579i)2-s − 0.986i·3-s + (0.993 − 0.115i)4-s − 1.52·5-s + (−0.0571 − 0.984i)6-s − 1.83i·7-s + (0.984 − 0.172i)8-s + 0.0277·9-s + (−1.52 + 0.0882i)10-s + 1.40i·11-s + (−0.114 − 0.979i)12-s + 1.40·13-s + (−0.106 − 1.83i)14-s + 1.50i·15-s + (0.973 − 0.229i)16-s − 0.483·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.115 + 0.993i$
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: \((2,\ 124,\ (\ :1),\ 0.115 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.58027 - 1.40699i\)
\(L(\frac12)\) \(\approx\) \(1.58027 - 1.40699i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 + 0.115i)T \)
31 \( 1 + 5.56iT \)
good3 \( 1 + 2.95iT - 9T^{2} \)
5 \( 1 + 7.62T + 25T^{2} \)
7 \( 1 + 12.8iT - 49T^{2} \)
11 \( 1 - 15.4iT - 121T^{2} \)
13 \( 1 - 18.2T + 169T^{2} \)
17 \( 1 + 8.21T + 289T^{2} \)
19 \( 1 - 13.1iT - 361T^{2} \)
23 \( 1 - 16.2iT - 529T^{2} \)
29 \( 1 + 5.38T + 841T^{2} \)
37 \( 1 - 23.5T + 1.36e3T^{2} \)
41 \( 1 + 18.0T + 1.68e3T^{2} \)
43 \( 1 - 15.1iT - 1.84e3T^{2} \)
47 \( 1 - 3.78iT - 2.20e3T^{2} \)
53 \( 1 - 32.3T + 2.80e3T^{2} \)
59 \( 1 - 47.8iT - 3.48e3T^{2} \)
61 \( 1 + 30.3T + 3.72e3T^{2} \)
67 \( 1 - 86.1iT - 4.48e3T^{2} \)
71 \( 1 + 100. iT - 5.04e3T^{2} \)
73 \( 1 + 12.7T + 5.32e3T^{2} \)
79 \( 1 - 81.8iT - 6.24e3T^{2} \)
83 \( 1 + 109. iT - 6.88e3T^{2} \)
89 \( 1 - 113.T + 7.92e3T^{2} \)
97 \( 1 + 76.4T + 9.40e3T^{2} \)
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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

−13.06330050034385862633075761662, −12.08640197176774070057305935471, −11.22149731279212858770949042579, −10.26856350719819142400632793825, −7.85595942585147729161226848374, −7.38977702270539876391388827835, −6.57050595215182840373682880081, −4.37024078357106476845643136479, −3.77740126566540322908809983573, −1.31638383305765574080661512625, 3.06236109045461070189904379415, 4.02998605272941081428760269623, 5.26518057119568784084399339771, 6.43009961910928583802879118292, 8.217236190250334839204181912767, 8.920851567864111917768697386440, 10.91294350389834344924688230000, 11.34381033070072560904061275852, 12.26067206994875438807229511405, 13.34809323827801206419071990932

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