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Properties

Label 2-112-112.19-c1-0-4
Degree $2$
Conductor $112$
Sign $0.993 - 0.112i$
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  + (−1.33 − 0.463i)2-s + (−0.0661 + 0.246i)3-s + (1.56 + 1.23i)4-s + (−0.499 + 0.133i)5-s + (0.202 − 0.299i)6-s + (2.45 + 0.998i)7-s + (−1.52 − 2.38i)8-s + (2.54 + 1.46i)9-s + (0.729 + 0.0528i)10-s + (−0.563 + 2.10i)11-s + (−0.409 + 0.305i)12-s + (4.30 − 4.30i)13-s + (−2.81 − 2.47i)14-s − 0.132i·15-s + (0.927 + 3.89i)16-s + (2.04 − 1.18i)17-s + ⋯
L(s)  = 1  + (−0.944 − 0.327i)2-s + (−0.0381 + 0.142i)3-s + (0.784 + 0.619i)4-s + (−0.223 + 0.0598i)5-s + (0.0828 − 0.122i)6-s + (0.926 + 0.377i)7-s + (−0.538 − 0.842i)8-s + (0.847 + 0.489i)9-s + (0.230 + 0.0167i)10-s + (−0.169 + 0.634i)11-s + (−0.118 + 0.0882i)12-s + (1.19 − 1.19i)13-s + (−0.751 − 0.660i)14-s − 0.0341i·15-s + (0.231 + 0.972i)16-s + (0.495 − 0.286i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.993 - 0.112i$
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: \((2,\ 112,\ (\ :1/2),\ 0.993 - 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759519 + 0.0429773i\)
\(L(\frac12)\) \(\approx\) \(0.759519 + 0.0429773i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.463i)T \)
7 \( 1 + (-2.45 - 0.998i)T \)
good3 \( 1 + (0.0661 - 0.246i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.499 - 0.133i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.563 - 2.10i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-4.30 + 4.30i)T - 13iT^{2} \)
17 \( 1 + (-2.04 + 1.18i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.07 - 1.35i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.78 - 3.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.02 + 5.02i)T + 29iT^{2} \)
31 \( 1 + (2.62 + 4.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.258 + 0.964i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.98T + 41T^{2} \)
43 \( 1 + (-2.39 - 2.39i)T + 43iT^{2} \)
47 \( 1 + (-4.72 + 8.17i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.557 + 0.149i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (9.38 + 2.51i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.948 + 3.54i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (5.02 + 1.34i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.49T + 71T^{2} \)
73 \( 1 + (4.24 + 7.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.71 + 0.988i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.925 - 0.925i)T + 83iT^{2} \)
89 \( 1 + (7.40 - 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.2iT - 97T^{2} \)
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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

−13.38143989983625290962862247592, −12.42449775901115907193191439566, −11.26695519302574139097710867780, −10.54211293392035388237947637416, −9.476719208095943134888344463245, −8.089579511086990414123097829627, −7.57592295132184029476462931587, −5.77026506562000188290276276557, −3.94384263475809262101186274097, −1.89450452524481684962074739661, 1.53999956482980380215640631443, 4.16084157025269060669534048200, 6.02879720436747195053115985838, 7.11704001086114068133252768249, 8.260269235655535406434901715358, 9.097287579647381335338746648399, 10.55150098313863054431749873377, 11.19441206240773003148787095492, 12.36542834270828669349860519791, 13.84762875577683328239981109726

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