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Properties

Label 2-124-124.123-c3-0-36
Degree $2$
Conductor $124$
Sign $0.839 - 0.542i$
Analytic cond. $7.31623$
Root an. cond. $2.70485$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.57 + 1.16i)2-s + 4.19·3-s + (5.27 + 6.01i)4-s + 15.8·5-s + (10.8 + 4.89i)6-s − 12.2i·7-s + (6.59 + 21.6i)8-s − 9.40·9-s + (40.7 + 18.4i)10-s − 58.7·11-s + (22.1 + 25.2i)12-s − 87.1i·13-s + (14.2 − 31.5i)14-s + 66.4·15-s + (−8.24 + 63.4i)16-s + 90.2i·17-s + ⋯
L(s)  = 1  + (0.911 + 0.412i)2-s + 0.807·3-s + (0.659 + 0.751i)4-s + 1.41·5-s + (0.735 + 0.332i)6-s − 0.661i·7-s + (0.291 + 0.956i)8-s − 0.348·9-s + (1.29 + 0.583i)10-s − 1.61·11-s + (0.532 + 0.606i)12-s − 1.85i·13-s + (0.272 − 0.602i)14-s + 1.14·15-s + (−0.128 + 0.991i)16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.839 - 0.542i$
Analytic conductor: \(7.31623\)
Root analytic conductor: \(2.70485\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :3/2),\ 0.839 - 0.542i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.63352 + 1.07147i\)
\(L(\frac12)\) \(\approx\) \(3.63352 + 1.07147i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.57 - 1.16i)T \)
31 \( 1 + (-25.3 + 170. i)T \)
good3 \( 1 - 4.19T + 27T^{2} \)
5 \( 1 - 15.8T + 125T^{2} \)
7 \( 1 + 12.2iT - 343T^{2} \)
11 \( 1 + 58.7T + 1.33e3T^{2} \)
13 \( 1 + 87.1iT - 2.19e3T^{2} \)
17 \( 1 - 90.2iT - 4.91e3T^{2} \)
19 \( 1 - 67.1iT - 6.85e3T^{2} \)
23 \( 1 - 29.5T + 1.21e4T^{2} \)
29 \( 1 - 51.5iT - 2.43e4T^{2} \)
37 \( 1 - 181. iT - 5.06e4T^{2} \)
41 \( 1 + 356.T + 6.89e4T^{2} \)
43 \( 1 - 430.T + 7.95e4T^{2} \)
47 \( 1 + 427. iT - 1.03e5T^{2} \)
53 \( 1 - 441. iT - 1.48e5T^{2} \)
59 \( 1 + 483. iT - 2.05e5T^{2} \)
61 \( 1 + 590. iT - 2.26e5T^{2} \)
67 \( 1 - 236. iT - 3.00e5T^{2} \)
71 \( 1 - 281. iT - 3.57e5T^{2} \)
73 \( 1 + 53.2iT - 3.89e5T^{2} \)
79 \( 1 - 637.T + 4.93e5T^{2} \)
83 \( 1 + 111.T + 5.71e5T^{2} \)
89 \( 1 + 226. iT - 7.04e5T^{2} \)
97 \( 1 + 67.1T + 9.12e5T^{2} \)
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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

−13.26325171884115930072847598024, −12.67949585783829103574875472344, −10.71091536286076465422744010178, −10.12926898313735031589586408046, −8.364729344783025837837157537305, −7.70662161185723517311949543057, −5.99434260653357594066969328576, −5.31180706989554817738080406202, −3.38169659117742787903306543986, −2.28519186566081340449548694891, 2.16698485386145377952142457408, 2.76810621166605324901561965182, 4.86953639028480725606627989049, 5.80605801253620719960419260619, 7.11594059590828177245113401973, 8.962468173779334259663855263747, 9.605891860736317640523701182807, 10.86293092823849943405564453768, 11.95857620240580178723662352755, 13.24082554728066791982937019694

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