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Properties

Label 2-112-28.27-c9-0-31
Degree $2$
Conductor $112$
Sign $-0.259 + 0.965i$
Analytic cond. $57.6840$
Root an. cond. $7.59499$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 168.·3-s − 1.69e3i·5-s + (6.13e3 − 1.64e3i)7-s + 8.85e3·9-s − 4.52e4i·11-s − 4.92e4i·13-s − 2.86e5i·15-s + 2.95e5i·17-s + 4.60e5·19-s + (1.03e6 − 2.77e5i)21-s − 1.79e6i·23-s − 9.15e5·25-s − 1.82e6·27-s − 5.43e6·29-s + 4.36e6·31-s + ⋯
L(s)  = 1  + 1.20·3-s − 1.21i·5-s + (0.966 − 0.258i)7-s + 0.449·9-s − 0.931i·11-s − 0.477i·13-s − 1.45i·15-s + 0.858i·17-s + 0.810·19-s + (1.16 − 0.311i)21-s − 1.33i·23-s − 0.468·25-s − 0.662·27-s − 1.42·29-s + 0.849·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.259 + 0.965i$
Analytic conductor: \(57.6840\)
Root analytic conductor: \(7.59499\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :9/2),\ -0.259 + 0.965i)\)

Particular Values

\(L(5)\) \(\approx\) \(3.321690622\)
\(L(\frac12)\) \(\approx\) \(3.321690622\)
\(L(\frac{11}{2})\) 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 \( 1 + (-6.13e3 + 1.64e3i)T \)
good3 \( 1 - 168.T + 1.96e4T^{2} \)
5 \( 1 + 1.69e3iT - 1.95e6T^{2} \)
11 \( 1 + 4.52e4iT - 2.35e9T^{2} \)
13 \( 1 + 4.92e4iT - 1.06e10T^{2} \)
17 \( 1 - 2.95e5iT - 1.18e11T^{2} \)
19 \( 1 - 4.60e5T + 3.22e11T^{2} \)
23 \( 1 + 1.79e6iT - 1.80e12T^{2} \)
29 \( 1 + 5.43e6T + 1.45e13T^{2} \)
31 \( 1 - 4.36e6T + 2.64e13T^{2} \)
37 \( 1 + 1.38e7T + 1.29e14T^{2} \)
41 \( 1 - 6.11e6iT - 3.27e14T^{2} \)
43 \( 1 - 1.02e7iT - 5.02e14T^{2} \)
47 \( 1 - 1.82e6T + 1.11e15T^{2} \)
53 \( 1 - 8.73e7T + 3.29e15T^{2} \)
59 \( 1 + 1.13e8T + 8.66e15T^{2} \)
61 \( 1 + 3.81e7iT - 1.16e16T^{2} \)
67 \( 1 - 2.09e8iT - 2.72e16T^{2} \)
71 \( 1 + 1.24e8iT - 4.58e16T^{2} \)
73 \( 1 + 2.09e8iT - 5.88e16T^{2} \)
79 \( 1 + 4.05e8iT - 1.19e17T^{2} \)
83 \( 1 + 1.21e8T + 1.86e17T^{2} \)
89 \( 1 + 1.05e9iT - 3.50e17T^{2} \)
97 \( 1 + 7.91e8iT - 7.60e17T^{2} \)
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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

−11.61032121144350789265808162299, −10.37434310614245396291797001713, −8.923708776712704576391351954275, −8.474857719580154840079553199784, −7.65375648423218011756974124994, −5.71306574630151163755735657829, −4.52730892507995152485848085924, −3.29669932712509982474218837345, −1.82691032429998699183075536502, −0.69310417546195827036836856259, 1.74340659175026725668089706585, 2.65383663711003301479652834327, 3.77001505549154399038824811023, 5.31450521188441031766397570030, 7.10806624880460599696992026998, 7.66564722165166718085813618798, 8.978231602208620568559189131663, 9.852623592189087550323131205057, 11.14739044147292312288073706987, 11.97219943795957508846789470866

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