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Properties

Label 2-123-41.9-c3-0-10
Degree $2$
Conductor $123$
Sign $-0.167 - 0.985i$
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  + 2.63i·2-s + (2.12 + 2.12i)3-s + 1.07·4-s − 9.61i·5-s + (−5.58 + 5.58i)6-s + (6.96 + 6.96i)7-s + 23.8i·8-s + 8.99i·9-s + 25.2·10-s + (20.2 + 20.2i)11-s + (2.28 + 2.28i)12-s + (28.8 + 28.8i)13-s + (−18.3 + 18.3i)14-s + (20.3 − 20.3i)15-s − 54.2·16-s + (4.20 − 4.20i)17-s + ⋯
L(s)  = 1  + 0.930i·2-s + (0.408 + 0.408i)3-s + 0.134·4-s − 0.859i·5-s + (−0.379 + 0.379i)6-s + (0.376 + 0.376i)7-s + 1.05i·8-s + 0.333i·9-s + 0.799·10-s + (0.555 + 0.555i)11-s + (0.0550 + 0.0550i)12-s + (0.616 + 0.616i)13-s + (−0.349 + 0.349i)14-s + (0.350 − 0.350i)15-s − 0.847·16-s + (0.0600 − 0.0600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(123\)    =    \(3 \cdot 41\)
Sign: $-0.167 - 0.985i$
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: \((2,\ 123,\ (\ :3/2),\ -0.167 - 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.39531 + 1.65213i\)
\(L(\frac12)\) \(\approx\) \(1.39531 + 1.65213i\)
\(L(\frac{5}{2})\) 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 \( 1 + (151. + 214. i)T \)
good2 \( 1 - 2.63iT - 8T^{2} \)
5 \( 1 + 9.61iT - 125T^{2} \)
7 \( 1 + (-6.96 - 6.96i)T + 343iT^{2} \)
11 \( 1 + (-20.2 - 20.2i)T + 1.33e3iT^{2} \)
13 \( 1 + (-28.8 - 28.8i)T + 2.19e3iT^{2} \)
17 \( 1 + (-4.20 + 4.20i)T - 4.91e3iT^{2} \)
19 \( 1 + (0.701 - 0.701i)T - 6.85e3iT^{2} \)
23 \( 1 + 128.T + 1.21e4T^{2} \)
29 \( 1 + (169. + 169. i)T + 2.43e4iT^{2} \)
31 \( 1 - 225.T + 2.97e4T^{2} \)
37 \( 1 - 78.4T + 5.06e4T^{2} \)
43 \( 1 - 59.3iT - 7.95e4T^{2} \)
47 \( 1 + (-77.4 + 77.4i)T - 1.03e5iT^{2} \)
53 \( 1 + (175. + 175. i)T + 1.48e5iT^{2} \)
59 \( 1 - 499.T + 2.05e5T^{2} \)
61 \( 1 + 297. iT - 2.26e5T^{2} \)
67 \( 1 + (-530. + 530. i)T - 3.00e5iT^{2} \)
71 \( 1 + (-129. - 129. i)T + 3.57e5iT^{2} \)
73 \( 1 - 318. iT - 3.89e5T^{2} \)
79 \( 1 + (396. + 396. i)T + 4.93e5iT^{2} \)
83 \( 1 - 8.78T + 5.71e5T^{2} \)
89 \( 1 + (608. + 608. i)T + 7.04e5iT^{2} \)
97 \( 1 + (1.05e3 - 1.05e3i)T - 9.12e5iT^{2} \)
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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

−13.55469374514927141751855439606, −12.14095401413536464938535511409, −11.31946178947158861992750769130, −9.770846681510945128094761209269, −8.701163238099842699649777998085, −7.986202904253592152164070456845, −6.60675209211150130551465634213, −5.38670620172108564196782032776, −4.19566964842940100800770117531, −1.98464356431036170422148100479, 1.25931017057559438469528527700, 2.80783323818071945800199479507, 3.84481228809896890655302734671, 6.15183932355890929240696965012, 7.16691017802720868628834076240, 8.371825051520023918616176183219, 9.830399394200909347174341383622, 10.79886366130684892441600557241, 11.46267831530899132477412711285, 12.54614557805016666717527883320

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