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Properties

Label 2-112-28.27-c7-0-11
Degree $2$
Conductor $112$
Sign $0.270 - 0.962i$
Analytic cond. $34.9871$
Root an. cond. $5.91499$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 67.4·3-s − 9.83i·5-s + (−245. + 873. i)7-s + 2.36e3·9-s − 2.41e3i·11-s + 8.44e3i·13-s − 663. i·15-s + 2.53e4i·17-s + 3.47e3·19-s + (−1.65e4 + 5.89e4i)21-s + 6.76e4i·23-s + 7.80e4·25-s + 1.21e4·27-s + 4.29e4·29-s + 4.78e4·31-s + ⋯
L(s)  = 1  + 1.44·3-s − 0.0351i·5-s + (−0.270 + 0.962i)7-s + 1.08·9-s − 0.546i·11-s + 1.06i·13-s − 0.0507i·15-s + 1.25i·17-s + 0.116·19-s + (−0.390 + 1.38i)21-s + 1.15i·23-s + 0.998·25-s + 0.118·27-s + 0.327·29-s + 0.288·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.270 - 0.962i$
Analytic conductor: \(34.9871\)
Root analytic conductor: \(5.91499\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :7/2),\ 0.270 - 0.962i)\)

Particular Values

\(L(4)\) \(\approx\) \(3.003081118\)
\(L(\frac12)\) \(\approx\) \(3.003081118\)
\(L(\frac{9}{2})\) 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 \( 1 + (245. - 873. i)T \)
good3 \( 1 - 67.4T + 2.18e3T^{2} \)
5 \( 1 + 9.83iT - 7.81e4T^{2} \)
11 \( 1 + 2.41e3iT - 1.94e7T^{2} \)
13 \( 1 - 8.44e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.53e4iT - 4.10e8T^{2} \)
19 \( 1 - 3.47e3T + 8.93e8T^{2} \)
23 \( 1 - 6.76e4iT - 3.40e9T^{2} \)
29 \( 1 - 4.29e4T + 1.72e10T^{2} \)
31 \( 1 - 4.78e4T + 2.75e10T^{2} \)
37 \( 1 + 3.54e5T + 9.49e10T^{2} \)
41 \( 1 + 5.54e5iT - 1.94e11T^{2} \)
43 \( 1 - 5.07e5iT - 2.71e11T^{2} \)
47 \( 1 - 6.35e5T + 5.06e11T^{2} \)
53 \( 1 - 1.26e6T + 1.17e12T^{2} \)
59 \( 1 - 9.21e5T + 2.48e12T^{2} \)
61 \( 1 - 1.86e6iT - 3.14e12T^{2} \)
67 \( 1 - 2.09e6iT - 6.06e12T^{2} \)
71 \( 1 - 4.07e6iT - 9.09e12T^{2} \)
73 \( 1 + 1.11e5iT - 1.10e13T^{2} \)
79 \( 1 + 2.72e5iT - 1.92e13T^{2} \)
83 \( 1 + 5.12e6T + 2.71e13T^{2} \)
89 \( 1 + 9.51e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.20e7iT - 8.07e13T^{2} \)
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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

−12.65091308452706657115675126533, −11.52769675980672281394837690753, −10.05740092679932190990573794566, −8.863050318480350837581275165327, −8.575633205013727822282032227501, −7.12626691016713535791021109877, −5.72059425794365059833086163558, −3.95222313963315610677066625302, −2.84403172619143634859845267717, −1.68351337561002437508110879108, 0.74066506571003786963471546989, 2.50490066053081337038524870069, 3.47521237166805949390120315003, 4.82908931540117572293404075746, 6.84066746168045157567512972571, 7.72143774730340426388005774176, 8.745177154011132720454989691116, 9.825749548920986506554728048668, 10.65898237763821093658136168076, 12.32072994456181145968058582567

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