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Properties

Label 2-112-28.3-c3-0-6
Degree $2$
Conductor $112$
Sign $0.994 - 0.104i$
Analytic cond. $6.60821$
Root an. cond. $2.57064$
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  + (−2.38 + 4.12i)3-s + (14.6 − 8.44i)5-s + (−8.89 − 16.2i)7-s + (2.13 + 3.69i)9-s + (35.2 + 20.3i)11-s − 56.7i·13-s + 80.5i·15-s + (106. + 61.4i)17-s + (37.7 + 65.3i)19-s + (88.2 + 1.97i)21-s + (91.9 − 53.0i)23-s + (80.2 − 138. i)25-s − 149.·27-s − 146.·29-s + (21.2 − 36.8i)31-s + ⋯
L(s)  = 1  + (−0.458 + 0.794i)3-s + (1.30 − 0.755i)5-s + (−0.480 − 0.876i)7-s + (0.0789 + 0.136i)9-s + (0.967 + 0.558i)11-s − 1.21i·13-s + 1.38i·15-s + (1.51 + 0.876i)17-s + (0.455 + 0.789i)19-s + (0.917 + 0.0204i)21-s + (0.833 − 0.481i)23-s + (0.641 − 1.11i)25-s − 1.06·27-s − 0.939·29-s + (0.123 − 0.213i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.994 - 0.104i$
Analytic conductor: \(6.60821\)
Root analytic conductor: \(2.57064\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :3/2),\ 0.994 - 0.104i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.73782 + 0.0908208i\)
\(L(\frac12)\) \(\approx\) \(1.73782 + 0.0908208i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (8.89 + 16.2i)T \)
good3 \( 1 + (2.38 - 4.12i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-14.6 + 8.44i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-35.2 - 20.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 56.7iT - 2.19e3T^{2} \)
17 \( 1 + (-106. - 61.4i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-37.7 - 65.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-91.9 + 53.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 146.T + 2.43e4T^{2} \)
31 \( 1 + (-21.2 + 36.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (40.4 + 70.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 53.8iT - 6.89e4T^{2} \)
43 \( 1 + 341. iT - 7.95e4T^{2} \)
47 \( 1 + (-2.06 - 3.57i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (139. - 242. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-87.4 + 151. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (404. - 233. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (652. + 376. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 669. iT - 3.57e5T^{2} \)
73 \( 1 + (-723. - 417. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (958. - 553. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 552.T + 5.71e5T^{2} \)
89 \( 1 + (-106. + 61.2i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 291. iT - 9.12e5T^{2} \)
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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

−13.09345572661978505878561918683, −12.33702657956167677512442085822, −10.60436795987137915155308029213, −10.06074510039019731086499895375, −9.296338975198574882123922804312, −7.65720162477918060881965817274, −6.02796724461790590403600647126, −5.15789641126635822558066077195, −3.74991766808278870770747950512, −1.31574599508135996475844914559, 1.47720248657550520003429829937, 3.06949773899024361862705190627, 5.52840126516581445268117479744, 6.39482998387857296461028469768, 7.12906852369799120333215703662, 9.221946944923387623207320500110, 9.642100779113206508225872612200, 11.34120687987426826329252088274, 12.00081184720662595834585789474, 13.19266861812602450849294514541

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