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Properties

Label 2-112-112.53-c1-0-10
Degree $2$
Conductor $112$
Sign $0.891 + 0.452i$
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  + (1.40 − 0.116i)2-s + (−0.312 − 1.16i)3-s + (1.97 − 0.327i)4-s + (−0.262 + 0.980i)5-s + (−0.575 − 1.60i)6-s + (−2.60 + 0.476i)7-s + (2.74 − 0.690i)8-s + (1.33 − 0.772i)9-s + (−0.256 + 1.41i)10-s + (−2.36 + 0.635i)11-s + (−0.997 − 2.19i)12-s + (−2.65 + 2.65i)13-s + (−3.61 + 0.973i)14-s + 1.22·15-s + (3.78 − 1.29i)16-s + (−0.509 + 0.881i)17-s + ⋯
L(s)  = 1  + (0.996 − 0.0820i)2-s + (−0.180 − 0.672i)3-s + (0.986 − 0.163i)4-s + (−0.117 + 0.438i)5-s + (−0.234 − 0.655i)6-s + (−0.983 + 0.180i)7-s + (0.969 − 0.243i)8-s + (0.445 − 0.257i)9-s + (−0.0811 + 0.446i)10-s + (−0.714 + 0.191i)11-s + (−0.287 − 0.634i)12-s + (−0.737 + 0.737i)13-s + (−0.965 + 0.260i)14-s + 0.316·15-s + (0.946 − 0.322i)16-s + (−0.123 + 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.891 + 0.452i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.891 + 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52931 - 0.366163i\)
\(L(\frac12)\) \(\approx\) \(1.52931 - 0.366163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.116i)T \)
7 \( 1 + (2.60 - 0.476i)T \)
good3 \( 1 + (0.312 + 1.16i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.262 - 0.980i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.36 - 0.635i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.65 - 2.65i)T - 13iT^{2} \)
17 \( 1 + (0.509 - 0.881i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0936 - 0.0250i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.67 - 0.965i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.05 - 5.05i)T - 29iT^{2} \)
31 \( 1 + (-4.28 + 7.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.06 + 7.71i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.51iT - 41T^{2} \)
43 \( 1 + (4.47 + 4.47i)T + 43iT^{2} \)
47 \( 1 + (-6.02 - 10.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.42 - 0.381i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-6.96 + 1.86i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.42 - 1.18i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.907 - 3.38i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.43iT - 71T^{2} \)
73 \( 1 + (7.34 + 4.23i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.433 + 0.751i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.44 - 5.44i)T - 83iT^{2} \)
89 \( 1 + (3.93 - 2.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−13.27476108548316796933733843851, −12.68051315904643553544640156104, −11.85723738535618645191406142030, −10.64125411295942174573931621562, −9.524007380585161954699348852841, −7.44256477035816122432856470963, −6.81873838565509818427084361049, −5.65514565704943448251077571952, −3.96521785984499075229076960226, −2.38927035280106154830518893846, 2.97131722450897549081785314961, 4.43407707930930086507539075627, 5.39783737208359744020745656380, 6.80603569914061527386206487490, 8.068688824272824504044498573220, 9.883233122699209571206041848645, 10.49964138050592124256186465356, 11.85488067040128426496562270576, 12.91467675356576867046317943582, 13.41438009641293340766452385914

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