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Properties

Label 2-112-28.3-c3-0-7
Degree $2$
Conductor $112$
Sign $0.587 + 0.809i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.587 + 0.809i$
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: \((2,\ 112,\ (\ :3/2),\ 0.587 + 0.809i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.01329 - 1.02619i\)
\(L(\frac12)\) \(\approx\) \(2.01329 - 1.02619i\)
\(L(\frac{5}{2})\) 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

−12.96066950763187395902626598161, −12.47653000897119007255596671173, −10.77498026702100515749591122039, −9.698434969955169580226348645796, −8.428141123340720748506781348826, −7.82148135058517877421491096062, −5.86933854883529379163938237145, −5.28445174234761030633303661409, −2.67766910274861851137094519320, −1.45767596953874969004765743320, 2.03462803012405506503048337690, 3.69204668226055400384794351819, 5.11226543764392466437999612235, 6.59357423502937833346123798191, 7.79383724826440442387199017803, 9.467320196450380102768700795717, 10.02689161311013491246236799720, 10.76378626432873113370734617107, 12.30402308920904456310118984007, 13.78422438932126305494588015146

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