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Properties

Label 2-112-112.19-c1-0-8
Degree $2$
Conductor $112$
Sign $0.990 - 0.136i$
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  + (−0.482 + 1.32i)2-s + (0.449 − 1.67i)3-s + (−1.53 − 1.28i)4-s + (0.731 − 0.195i)5-s + (2.01 + 1.40i)6-s + (2.52 + 0.800i)7-s + (2.44 − 1.42i)8-s + (−0.0183 − 0.0105i)9-s + (−0.0923 + 1.06i)10-s + (1.18 − 4.42i)11-s + (−2.84 + 1.99i)12-s + (−2.89 + 2.89i)13-s + (−2.28 + 2.96i)14-s − 1.31i·15-s + (0.708 + 3.93i)16-s + (−2.28 + 1.31i)17-s + ⋯
L(s)  = 1  + (−0.341 + 0.940i)2-s + (0.259 − 0.969i)3-s + (−0.767 − 0.641i)4-s + (0.326 − 0.0875i)5-s + (0.822 + 0.574i)6-s + (0.953 + 0.302i)7-s + (0.864 − 0.502i)8-s + (−0.00611 − 0.00353i)9-s + (−0.0291 + 0.337i)10-s + (0.357 − 1.33i)11-s + (−0.820 + 0.577i)12-s + (−0.803 + 0.803i)13-s + (−0.609 + 0.792i)14-s − 0.339i·15-s + (0.177 + 0.984i)16-s + (−0.554 + 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.990 - 0.136i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.990 - 0.136i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.993554 + 0.0679694i\)
\(L(\frac12)\) \(\approx\) \(0.993554 + 0.0679694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.482 - 1.32i)T \)
7 \( 1 + (-2.52 - 0.800i)T \)
good3 \( 1 + (-0.449 + 1.67i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.731 + 0.195i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.18 + 4.42i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.89 - 2.89i)T - 13iT^{2} \)
17 \( 1 + (2.28 - 1.31i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.38 - 1.44i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.01 + 1.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.209 + 0.209i)T + 29iT^{2} \)
31 \( 1 + (-3.33 - 5.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.02 - 3.82i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.04T + 41T^{2} \)
43 \( 1 + (3.79 + 3.79i)T + 43iT^{2} \)
47 \( 1 + (2.53 - 4.39i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.7 - 2.87i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.23 + 1.40i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.56 + 5.84i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-9.24 - 2.47i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.25T + 71T^{2} \)
73 \( 1 + (-3.29 - 5.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.0 + 7.54i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.00 + 8.00i)T + 83iT^{2} \)
89 \( 1 + (-3.92 + 6.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.79iT - 97T^{2} \)
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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

−13.88304694103452157484914954930, −12.96415269225537393741011991479, −11.65517850024973222080049111836, −10.34115092030571815503653137687, −8.806922147966368396971997761440, −8.257902015416102658496114809587, −7.02120377473140807609128150031, −6.05511748360815698641777961540, −4.59946557186030420990163887564, −1.75239762659366014216310199979, 2.21974648510676056915956171564, 4.12438827462430677204807876988, 4.88465947833092085344389568133, 7.28890835559451645339350488055, 8.562043735139347604299241774608, 9.749369833029565864481409269276, 10.21631948264173061057405772801, 11.31752750835480677232718614508, 12.41634242559071530528600291811, 13.48414146867245101854223621190

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