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Properties

Label 2-124-124.123-c3-0-21
Degree $2$
Conductor $124$
Sign $0.686 - 0.727i$
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  + (−2.59 + 1.11i)2-s + 8.21·3-s + (5.49 − 5.81i)4-s + 6.37·5-s + (−21.3 + 9.19i)6-s + 24.3i·7-s + (−7.78 + 21.2i)8-s + 40.5·9-s + (−16.5 + 7.12i)10-s + 4.11·11-s + (45.1 − 47.7i)12-s − 3.21i·13-s + (−27.2 − 63.3i)14-s + 52.3·15-s + (−3.53 − 63.9i)16-s + 7.66i·17-s + ⋯
L(s)  = 1  + (−0.918 + 0.395i)2-s + 1.58·3-s + (0.687 − 0.726i)4-s + 0.569·5-s + (−1.45 + 0.625i)6-s + 1.31i·7-s + (−0.344 + 0.938i)8-s + 1.50·9-s + (−0.523 + 0.225i)10-s + 0.112·11-s + (1.08 − 1.14i)12-s − 0.0686i·13-s + (−0.520 − 1.20i)14-s + 0.901·15-s + (−0.0551 − 0.998i)16-s + 0.109i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.686 - 0.727i$
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: \((2,\ 124,\ (\ :3/2),\ 0.686 - 0.727i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.79886 + 0.776013i\)
\(L(\frac12)\) \(\approx\) \(1.79886 + 0.776013i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.59 - 1.11i)T \)
31 \( 1 + (-172. + 0.266i)T \)
good3 \( 1 - 8.21T + 27T^{2} \)
5 \( 1 - 6.37T + 125T^{2} \)
7 \( 1 - 24.3iT - 343T^{2} \)
11 \( 1 - 4.11T + 1.33e3T^{2} \)
13 \( 1 + 3.21iT - 2.19e3T^{2} \)
17 \( 1 - 7.66iT - 4.91e3T^{2} \)
19 \( 1 + 53.5iT - 6.85e3T^{2} \)
23 \( 1 - 134.T + 1.21e4T^{2} \)
29 \( 1 - 191. iT - 2.43e4T^{2} \)
37 \( 1 + 342. iT - 5.06e4T^{2} \)
41 \( 1 + 368.T + 6.89e4T^{2} \)
43 \( 1 - 0.524T + 7.95e4T^{2} \)
47 \( 1 + 521. iT - 1.03e5T^{2} \)
53 \( 1 - 679. iT - 1.48e5T^{2} \)
59 \( 1 - 151. iT - 2.05e5T^{2} \)
61 \( 1 + 622. iT - 2.26e5T^{2} \)
67 \( 1 + 639. iT - 3.00e5T^{2} \)
71 \( 1 + 406. iT - 3.57e5T^{2} \)
73 \( 1 + 424. iT - 3.89e5T^{2} \)
79 \( 1 + 985.T + 4.93e5T^{2} \)
83 \( 1 - 811.T + 5.71e5T^{2} \)
89 \( 1 - 729. iT - 7.04e5T^{2} \)
97 \( 1 + 453.T + 9.12e5T^{2} \)
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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

−13.35383131304536534653624352975, −12.03964707813040486263819700636, −10.57241172552002622397487215025, −9.259537749013302488410319888505, −9.028249502668560966985308980853, −8.023011266974462747349882469610, −6.77132007416026662933027034138, −5.35215661878032486155963772505, −2.93349768185467870695785320092, −1.89848091843810202745716900664, 1.38427117016262467890074606560, 2.85010608261002476004543363935, 4.05238955010947491754452442450, 6.71022126744212362030766858487, 7.74519455904033961360359472115, 8.547134770157067053750734395987, 9.757066678151253533110058806315, 10.15307794128475540405595825855, 11.57222298331816250174953532994, 13.15536043735078610825496710312

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