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Properties

Label 2-123-41.9-c3-0-9
Degree $2$
Conductor $123$
Sign $0.0213 - 0.999i$
Analytic cond. $7.25723$
Root an. cond. $2.69392$
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  + 0.702i·2-s + (2.12 + 2.12i)3-s + 7.50·4-s + 13.0i·5-s + (−1.49 + 1.49i)6-s + (4.57 + 4.57i)7-s + 10.8i·8-s + 8.99i·9-s − 9.19·10-s + (−40.8 − 40.8i)11-s + (15.9 + 15.9i)12-s + (26.3 + 26.3i)13-s + (−3.21 + 3.21i)14-s + (−27.7 + 27.7i)15-s + 52.3·16-s + (−9.57 + 9.57i)17-s + ⋯
L(s)  = 1  + 0.248i·2-s + (0.408 + 0.408i)3-s + 0.938·4-s + 1.17i·5-s + (−0.101 + 0.101i)6-s + (0.247 + 0.247i)7-s + 0.481i·8-s + 0.333i·9-s − 0.290·10-s + (−1.11 − 1.11i)11-s + (0.383 + 0.383i)12-s + (0.562 + 0.562i)13-s + (−0.0613 + 0.0613i)14-s + (−0.478 + 0.478i)15-s + 0.818·16-s + (−0.136 + 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0213 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0213 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(123\)    =    \(3 \cdot 41\)
Sign: $0.0213 - 0.999i$
Analytic conductor: \(7.25723\)
Root analytic conductor: \(2.69392\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{123} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 123,\ (\ :3/2),\ 0.0213 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.56801 + 1.53483i\)
\(L(\frac12)\) \(\approx\) \(1.56801 + 1.53483i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.12 - 2.12i)T \)
41 \( 1 + (108. + 238. i)T \)
good2 \( 1 - 0.702iT - 8T^{2} \)
5 \( 1 - 13.0iT - 125T^{2} \)
7 \( 1 + (-4.57 - 4.57i)T + 343iT^{2} \)
11 \( 1 + (40.8 + 40.8i)T + 1.33e3iT^{2} \)
13 \( 1 + (-26.3 - 26.3i)T + 2.19e3iT^{2} \)
17 \( 1 + (9.57 - 9.57i)T - 4.91e3iT^{2} \)
19 \( 1 + (-13.5 + 13.5i)T - 6.85e3iT^{2} \)
23 \( 1 + 47.2T + 1.21e4T^{2} \)
29 \( 1 + (-150. - 150. i)T + 2.43e4iT^{2} \)
31 \( 1 + 130.T + 2.97e4T^{2} \)
37 \( 1 - 324.T + 5.06e4T^{2} \)
43 \( 1 + 283. iT - 7.95e4T^{2} \)
47 \( 1 + (-324. + 324. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-231. - 231. i)T + 1.48e5iT^{2} \)
59 \( 1 + 337.T + 2.05e5T^{2} \)
61 \( 1 + 688. iT - 2.26e5T^{2} \)
67 \( 1 + (429. - 429. i)T - 3.00e5iT^{2} \)
71 \( 1 + (390. + 390. i)T + 3.57e5iT^{2} \)
73 \( 1 + 675. iT - 3.89e5T^{2} \)
79 \( 1 + (-539. - 539. i)T + 4.93e5iT^{2} \)
83 \( 1 - 72.0T + 5.71e5T^{2} \)
89 \( 1 + (-574. - 574. i)T + 7.04e5iT^{2} \)
97 \( 1 + (-262. + 262. i)T - 9.12e5iT^{2} \)
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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

−13.52413318732812221426501581598, −11.90604243619875015834543799737, −10.81936014137306789909275631938, −10.53923397560515534917276211068, −8.779855634437026711465158757161, −7.72936408424173307660658442445, −6.63083691667581977535233627952, −5.51000517999024601157725603393, −3.41770277354018705875903946331, −2.36782679166308180956775331256, 1.19643004413255582048937042620, 2.62798758625528994161629721964, 4.50501108402762098522909670680, 5.96954447959542514125667330829, 7.49588580194746573836564983940, 8.148846849172789860099553680729, 9.597006889095826479988389746786, 10.63372374718705688261864566223, 11.87340182045899935871934537694, 12.73109614400017213906979336619

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