Properties

Label 2-112-4.3-c4-0-7
Degree $2$
Conductor $112$
Sign $0.866 + 0.5i$
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  − 6.40i·3-s + 38.0·5-s + 18.5i·7-s + 39.9·9-s + 70.7i·11-s − 59.7·13-s − 243. i·15-s + 359.·17-s + 24.0i·19-s + 118.·21-s − 985. i·23-s + 820.·25-s − 774. i·27-s + 1.11e3·29-s + 656. i·31-s + ⋯
L(s)  = 1  − 0.711i·3-s + 1.52·5-s + 0.377i·7-s + 0.493·9-s + 0.584i·11-s − 0.353·13-s − 1.08i·15-s + 1.24·17-s + 0.0664i·19-s + 0.269·21-s − 1.86i·23-s + 1.31·25-s − 1.06i·27-s + 1.32·29-s + 0.683i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(11.5774\)
Root analytic conductor: \(3.40256\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :2),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.29328 - 0.614484i\)
\(L(\frac12)\) \(\approx\) \(2.29328 - 0.614484i\)
\(L(3)\) 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 \( 1 - 18.5iT \)
good3 \( 1 + 6.40iT - 81T^{2} \)
5 \( 1 - 38.0T + 625T^{2} \)
11 \( 1 - 70.7iT - 1.46e4T^{2} \)
13 \( 1 + 59.7T + 2.85e4T^{2} \)
17 \( 1 - 359.T + 8.35e4T^{2} \)
19 \( 1 - 24.0iT - 1.30e5T^{2} \)
23 \( 1 + 985. iT - 2.79e5T^{2} \)
29 \( 1 - 1.11e3T + 7.07e5T^{2} \)
31 \( 1 - 656. iT - 9.23e5T^{2} \)
37 \( 1 + 1.92e3T + 1.87e6T^{2} \)
41 \( 1 + 304.T + 2.82e6T^{2} \)
43 \( 1 + 1.21e3iT - 3.41e6T^{2} \)
47 \( 1 - 3.31e3iT - 4.87e6T^{2} \)
53 \( 1 - 57.6T + 7.89e6T^{2} \)
59 \( 1 + 4.47e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.63e3T + 1.38e7T^{2} \)
67 \( 1 - 2.60e3iT - 2.01e7T^{2} \)
71 \( 1 - 4.94e3iT - 2.54e7T^{2} \)
73 \( 1 + 7.89e3T + 2.83e7T^{2} \)
79 \( 1 - 9.56e3iT - 3.89e7T^{2} \)
83 \( 1 - 5.92e3iT - 4.74e7T^{2} \)
89 \( 1 + 4.47e3T + 6.27e7T^{2} \)
97 \( 1 - 1.58e4T + 8.85e7T^{2} \)
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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

−12.61617395700959228555626989244, −12.32596075748781600560189193959, −10.38783963896545365328566604630, −9.806994880735296949132243295703, −8.499340915191872362439819108809, −7.06595157629523807428336260952, −6.14480596155045786510857467568, −4.88734853034560212540703613161, −2.55122830176367713256311980552, −1.37072429663287437397662562212, 1.48139320946916739215590598300, 3.34301926756322860235705931173, 4.99455657785465050773522267191, 5.98545067072936084656715873980, 7.42102094686300815228492278709, 9.056290530821660942547673498943, 9.970760185952206732981365063439, 10.43725966166206873717761315250, 11.93693768669022769293058841059, 13.31469215394056087948563044607

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