Properties

Label 2-112-28.23-c4-0-14
Degree $2$
Conductor $112$
Sign $-0.0595 + 0.998i$
Analytic cond. $11.5774$
Root an. cond. $3.40256$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.49 − 3.74i)3-s + (24.4 − 42.4i)5-s + (29.5 − 39.1i)7-s + (−12.3 + 21.4i)9-s + (−69.7 + 40.2i)11-s − 194.·13-s − 367. i·15-s + (−58.6 − 101. i)17-s + (486. + 280. i)19-s + (45.0 − 364. i)21-s + (149. + 86.3i)23-s + (−887. − 1.53e3i)25-s + 793. i·27-s + 156.·29-s + (1.26e3 − 732. i)31-s + ⋯
L(s)  = 1  + (0.721 − 0.416i)3-s + (0.979 − 1.69i)5-s + (0.602 − 0.798i)7-s + (−0.152 + 0.264i)9-s + (−0.576 + 0.332i)11-s − 1.15·13-s − 1.63i·15-s + (−0.203 − 0.351i)17-s + (1.34 + 0.777i)19-s + (0.102 − 0.826i)21-s + (0.282 + 0.163i)23-s + (−1.42 − 2.45i)25-s + 1.08i·27-s + 0.186·29-s + (1.31 − 0.761i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0595 + 0.998i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0595 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.0595 + 0.998i$
Analytic conductor: \(11.5774\)
Root analytic conductor: \(3.40256\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :2),\ -0.0595 + 0.998i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.70024 - 1.80465i\)
\(L(\frac12)\) \(\approx\) \(1.70024 - 1.80465i\)
\(L(3)\) 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 \( 1 + (-29.5 + 39.1i)T \)
good3 \( 1 + (-6.49 + 3.74i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-24.4 + 42.4i)T + (-312.5 - 541. i)T^{2} \)
11 \( 1 + (69.7 - 40.2i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 194.T + 2.85e4T^{2} \)
17 \( 1 + (58.6 + 101. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-486. - 280. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-149. - 86.3i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 - 156.T + 7.07e5T^{2} \)
31 \( 1 + (-1.26e3 + 732. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (347. - 602. i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 618.T + 2.82e6T^{2} \)
43 \( 1 + 1.55e3iT - 3.41e6T^{2} \)
47 \( 1 + (-2.22e3 - 1.28e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-788. - 1.36e3i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (4.32e3 - 2.49e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-800. + 1.38e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.25e3 + 1.87e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 4.04e3iT - 2.54e7T^{2} \)
73 \( 1 + (-3.63e3 - 6.29e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (234. + 135. i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 3.46e3iT - 4.74e7T^{2} \)
89 \( 1 + (-4.86e3 + 8.41e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 8.06e3T + 8.85e7T^{2} \)
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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

−12.86138071821876293785176489977, −11.91046679698594704229071419867, −10.20086545216328431971344444327, −9.365522396989568963516880785465, −8.214679090181509175544383242416, −7.46554349379722237451436831105, −5.44122512031786641082727892475, −4.63373597487137529231707900776, −2.33436941656705077155323394534, −1.06550912348751916736591523583, 2.42040771066718818418889527728, 3.06484966956407310794143591302, 5.21218607404252103326441396898, 6.45566145756047143685545283516, 7.70033620941003914518854466896, 9.106595856059941444086812731268, 9.931390897666168199154485475963, 10.90978604416085627815568318881, 11.99271760522638354802654326383, 13.63181480911160737775530168915

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