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Properties

Label 2-124-4.3-c2-0-4
Degree $2$
Conductor $124$
Sign $-0.918 - 0.394i$
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  + (−1.66 + 1.10i)2-s + 3.18i·3-s + (1.57 − 3.67i)4-s + 1.05·5-s + (−3.50 − 5.31i)6-s + 9.87i·7-s + (1.41 + 7.87i)8-s − 1.12·9-s + (−1.76 + 1.16i)10-s + 0.537i·11-s + (11.6 + 5.02i)12-s − 19.7·13-s + (−10.8 − 16.4i)14-s + 3.37i·15-s + (−11.0 − 11.5i)16-s − 3.48·17-s + ⋯
L(s)  = 1  + (−0.834 + 0.550i)2-s + 1.06i·3-s + (0.394 − 0.918i)4-s + 0.211·5-s + (−0.583 − 0.885i)6-s + 1.41i·7-s + (0.176 + 0.984i)8-s − 0.125·9-s + (−0.176 + 0.116i)10-s + 0.0489i·11-s + (0.974 + 0.418i)12-s − 1.51·13-s + (−0.776 − 1.17i)14-s + 0.224i·15-s + (−0.688 − 0.724i)16-s − 0.205·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.918 - 0.394i$
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: \((2,\ 124,\ (\ :1),\ -0.918 - 0.394i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.166107 + 0.808199i\)
\(L(\frac12)\) \(\approx\) \(0.166107 + 0.808199i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.66 - 1.10i)T \)
31 \( 1 - 5.56iT \)
good3 \( 1 - 3.18iT - 9T^{2} \)
5 \( 1 - 1.05T + 25T^{2} \)
7 \( 1 - 9.87iT - 49T^{2} \)
11 \( 1 - 0.537iT - 121T^{2} \)
13 \( 1 + 19.7T + 169T^{2} \)
17 \( 1 + 3.48T + 289T^{2} \)
19 \( 1 + 11.6iT - 361T^{2} \)
23 \( 1 - 25.0iT - 529T^{2} \)
29 \( 1 - 50.7T + 841T^{2} \)
37 \( 1 - 2.98T + 1.36e3T^{2} \)
41 \( 1 - 36.6T + 1.68e3T^{2} \)
43 \( 1 + 17.9iT - 1.84e3T^{2} \)
47 \( 1 - 27.7iT - 2.20e3T^{2} \)
53 \( 1 - 73.0T + 2.80e3T^{2} \)
59 \( 1 - 23.9iT - 3.48e3T^{2} \)
61 \( 1 - 84.2T + 3.72e3T^{2} \)
67 \( 1 + 19.5iT - 4.48e3T^{2} \)
71 \( 1 + 56.5iT - 5.04e3T^{2} \)
73 \( 1 - 52.4T + 5.32e3T^{2} \)
79 \( 1 - 122. iT - 6.24e3T^{2} \)
83 \( 1 + 42.9iT - 6.88e3T^{2} \)
89 \( 1 + 108.T + 7.92e3T^{2} \)
97 \( 1 + 99.6T + 9.40e3T^{2} \)
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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

−14.02875242927423451224732686947, −12.33198035674908857087457987415, −11.31660096000396038702261781544, −10.00823094239824626302025949084, −9.517128203974338073204393363879, −8.579540364729406855715420237717, −7.18818904055360048266670546070, −5.71518169245657125095002462674, −4.80090971376824780190061962788, −2.45172759073286637453496311146, 0.74926113098044120242368045168, 2.33514642461732094405407946416, 4.25461267275253070751214555869, 6.62832988394250003437075966323, 7.37270142273812630813539133150, 8.212886322570607787707626059869, 9.851274805175771277042491266922, 10.41509064194892978184226390843, 11.79894593327014655628128162008, 12.58861619338456593549961628669

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