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Properties

Label 2-112-16.13-c1-0-2
Degree $2$
Conductor $112$
Sign $0.142 - 0.989i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.857 + 1.12i)2-s + (0.416 + 0.416i)3-s + (−0.529 + 1.92i)4-s + (−1.13 + 1.13i)5-s + (−0.111 + 0.826i)6-s − i·7-s + (−2.62 + 1.05i)8-s − 2.65i·9-s + (−2.24 − 0.302i)10-s + (3.85 − 3.85i)11-s + (−1.02 + 0.583i)12-s + (4.66 + 4.66i)13-s + (1.12 − 0.857i)14-s − 0.943·15-s + (−3.43 − 2.04i)16-s − 5.33·17-s + ⋯
L(s)  = 1  + (0.606 + 0.795i)2-s + (0.240 + 0.240i)3-s + (−0.264 + 0.964i)4-s + (−0.506 + 0.506i)5-s + (−0.0454 + 0.337i)6-s − 0.377i·7-s + (−0.927 + 0.374i)8-s − 0.884i·9-s + (−0.709 − 0.0956i)10-s + (1.16 − 1.16i)11-s + (−0.295 + 0.168i)12-s + (1.29 + 1.29i)13-s + (0.300 − 0.229i)14-s − 0.243·15-s + (−0.859 − 0.510i)16-s − 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.142 - 0.989i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.142 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02595 + 0.888485i\)
\(L(\frac12)\) \(\approx\) \(1.02595 + 0.888485i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.857 - 1.12i)T \)
7 \( 1 + iT \)
good3 \( 1 + (-0.416 - 0.416i)T + 3iT^{2} \)
5 \( 1 + (1.13 - 1.13i)T - 5iT^{2} \)
11 \( 1 + (-3.85 + 3.85i)T - 11iT^{2} \)
13 \( 1 + (-4.66 - 4.66i)T + 13iT^{2} \)
17 \( 1 + 5.33T + 17T^{2} \)
19 \( 1 + (2.55 + 2.55i)T + 19iT^{2} \)
23 \( 1 + 2.60iT - 23T^{2} \)
29 \( 1 + (1.22 + 1.22i)T + 29iT^{2} \)
31 \( 1 + 0.833T + 31T^{2} \)
37 \( 1 + (4.42 - 4.42i)T - 37iT^{2} \)
41 \( 1 + 0.263iT - 41T^{2} \)
43 \( 1 + (-1.25 + 1.25i)T - 43iT^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + (-0.0476 + 0.0476i)T - 53iT^{2} \)
59 \( 1 + (3.60 - 3.60i)T - 59iT^{2} \)
61 \( 1 + (-4.46 - 4.46i)T + 61iT^{2} \)
67 \( 1 + (-9.50 - 9.50i)T + 67iT^{2} \)
71 \( 1 - 2.05iT - 71T^{2} \)
73 \( 1 + 5.48iT - 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 + (-5.84 - 5.84i)T + 83iT^{2} \)
89 \( 1 + 6.32iT - 89T^{2} \)
97 \( 1 - 18.8T + 97T^{2} \)
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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

−14.01618041603207926317015133427, −13.21764412381449056448609424130, −11.66980438585559052514964329201, −11.16445462044163673799893426647, −9.076881006238932017436568111526, −8.574879337451563469951014186223, −6.75255886888643871248670666024, −6.39736777869086950574651055015, −4.22096134465903823923396071398, −3.51202375648592624017525777590, 1.87548287171418187223294974399, 3.77664761717833732516716301428, 4.97758784519603427389361376926, 6.44155736508369976520404556459, 8.148376754472034342042670139343, 9.149853285280924040486922856000, 10.51060722916102380415294878314, 11.43362650451912880452235458205, 12.55991534309765989034827482067, 13.08936769140933714328680673987

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