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Properties

Label 2-124-124.123-c3-0-33
Degree $2$
Conductor $124$
Sign $-0.0438 + 0.999i$
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  + (−1.63 + 2.30i)2-s − 0.254·3-s + (−2.63 − 7.55i)4-s + 3.31·5-s + (0.416 − 0.586i)6-s + 7.54i·7-s + (21.7 + 6.29i)8-s − 26.9·9-s + (−5.42 + 7.63i)10-s − 43.9·11-s + (0.670 + 1.92i)12-s − 69.9i·13-s + (−17.4 − 12.3i)14-s − 0.842·15-s + (−50.1 + 39.8i)16-s − 85.3i·17-s + ⋯
L(s)  = 1  + (−0.579 + 0.815i)2-s − 0.0489·3-s + (−0.329 − 0.944i)4-s + 0.296·5-s + (0.0283 − 0.0399i)6-s + 0.407i·7-s + (0.960 + 0.278i)8-s − 0.997·9-s + (−0.171 + 0.241i)10-s − 1.20·11-s + (0.0161 + 0.0462i)12-s − 1.49i·13-s + (−0.332 − 0.235i)14-s − 0.0144·15-s + (−0.782 + 0.622i)16-s − 1.21i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0438 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0438 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.0438 + 0.999i$
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: \((2,\ 124,\ (\ :3/2),\ -0.0438 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.292042 - 0.305153i\)
\(L(\frac12)\) \(\approx\) \(0.292042 - 0.305153i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.63 - 2.30i)T \)
31 \( 1 + (160. + 63.9i)T \)
good3 \( 1 + 0.254T + 27T^{2} \)
5 \( 1 - 3.31T + 125T^{2} \)
7 \( 1 - 7.54iT - 343T^{2} \)
11 \( 1 + 43.9T + 1.33e3T^{2} \)
13 \( 1 + 69.9iT - 2.19e3T^{2} \)
17 \( 1 + 85.3iT - 4.91e3T^{2} \)
19 \( 1 - 8.51iT - 6.85e3T^{2} \)
23 \( 1 - 129.T + 1.21e4T^{2} \)
29 \( 1 + 100. iT - 2.43e4T^{2} \)
37 \( 1 + 297. iT - 5.06e4T^{2} \)
41 \( 1 + 134.T + 6.89e4T^{2} \)
43 \( 1 + 495.T + 7.95e4T^{2} \)
47 \( 1 - 440. iT - 1.03e5T^{2} \)
53 \( 1 - 174. iT - 1.48e5T^{2} \)
59 \( 1 + 365. iT - 2.05e5T^{2} \)
61 \( 1 - 608. iT - 2.26e5T^{2} \)
67 \( 1 - 790. iT - 3.00e5T^{2} \)
71 \( 1 + 586. iT - 3.57e5T^{2} \)
73 \( 1 - 771. iT - 3.89e5T^{2} \)
79 \( 1 + 307.T + 4.93e5T^{2} \)
83 \( 1 - 862.T + 5.71e5T^{2} \)
89 \( 1 + 413. iT - 7.04e5T^{2} \)
97 \( 1 + 1.38e3T + 9.12e5T^{2} \)
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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

−12.94334100880067546957819203341, −11.37074950881760802691313393879, −10.40236701421737731380238314205, −9.343392828528421033688146483094, −8.275898542947880391916342574308, −7.38710575182719951870412355087, −5.74716211358626576322780127656, −5.25284595728070649520432024273, −2.69587439747084926641964549595, −0.25369973245687365485610314546, 1.90410933464965567465112292470, 3.44333072518822257359609062843, 5.05265381252518007807390741592, 6.78400111624761009859401753212, 8.149118367006767863847788766070, 9.023172456335458612283955688048, 10.21839887034477226197616200503, 11.03463827916121640261775596809, 11.94497795722584444086217195987, 13.15922470449627418350257734565

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