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Properties

Label 2-112-112.109-c1-0-10
Degree $2$
Conductor $112$
Sign $0.987 + 0.160i$
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  + (1.41 − 0.0428i)2-s + (0.831 − 0.222i)3-s + (1.99 − 0.121i)4-s + (−2.02 − 0.543i)5-s + (1.16 − 0.350i)6-s + (−2.63 − 0.233i)7-s + (2.81 − 0.256i)8-s + (−1.95 + 1.12i)9-s + (−2.88 − 0.681i)10-s + (1.03 + 3.85i)11-s + (1.63 − 0.545i)12-s + (−0.990 − 0.990i)13-s + (−3.73 − 0.216i)14-s − 1.80·15-s + (3.97 − 0.483i)16-s + (3.07 − 5.33i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0303i)2-s + (0.480 − 0.128i)3-s + (0.998 − 0.0605i)4-s + (−0.906 − 0.242i)5-s + (0.476 − 0.143i)6-s + (−0.996 − 0.0881i)7-s + (0.995 − 0.0907i)8-s + (−0.652 + 0.376i)9-s + (−0.913 − 0.215i)10-s + (0.311 + 1.16i)11-s + (0.471 − 0.157i)12-s + (−0.274 − 0.274i)13-s + (−0.998 − 0.0579i)14-s − 0.466·15-s + (0.992 − 0.120i)16-s + (0.746 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.987 + 0.160i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.987 + 0.160i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67606 - 0.135520i\)
\(L(\frac12)\) \(\approx\) \(1.67606 - 0.135520i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0428i)T \)
7 \( 1 + (2.63 + 0.233i)T \)
good3 \( 1 + (-0.831 + 0.222i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (2.02 + 0.543i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.03 - 3.85i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.990 + 0.990i)T + 13iT^{2} \)
17 \( 1 + (-3.07 + 5.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.01 - 3.79i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.91 + 3.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.83 + 3.83i)T + 29iT^{2} \)
31 \( 1 + (2.05 - 3.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0740 - 0.0198i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.68iT - 41T^{2} \)
43 \( 1 + (-0.713 + 0.713i)T - 43iT^{2} \)
47 \( 1 + (-1.95 - 3.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.89 - 7.06i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.851 + 3.17i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.37 - 8.84i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.49 - 0.401i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.86iT - 71T^{2} \)
73 \( 1 + (-8.95 - 5.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.33 + 5.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.2 - 10.2i)T + 83iT^{2} \)
89 \( 1 + (1.16 - 0.671i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 18.7T + 97T^{2} \)
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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

−13.61196765047574921778512831694, −12.48690574467080601496024624648, −12.00988336060552815401545267742, −10.66163278936194834266942047424, −9.369454653282047032140553325795, −7.79339512576288402726924570967, −6.99996653273255337129993257539, −5.39633796738455994524373375599, −3.99398810871188323944586383897, −2.73685861755943070441017125996, 3.12885775793798443163381927323, 3.75704324079825771597658890748, 5.68878777495598799961853375706, 6.78437692109769067320267831178, 8.076207917308130160078641360692, 9.323370781573329987069825100154, 10.93879478004517633909319485125, 11.64961496926830586694324748852, 12.77366899133456947417423073242, 13.63840408461535393344539152937

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