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Properties

Label 2-112-112.93-c1-0-8
Degree $2$
Conductor $112$
Sign $0.156 + 0.987i$
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 − 0.00789i)2-s + (0.814 − 3.04i)3-s + (1.99 + 0.0223i)4-s + (0.501 + 1.87i)5-s + (−1.17 + 4.29i)6-s + (1.89 − 1.84i)7-s + (−2.82 − 0.0473i)8-s + (−5.98 − 3.45i)9-s + (−0.694 − 2.65i)10-s + (−1.11 − 0.299i)11-s + (1.69 − 6.06i)12-s + (0.00680 + 0.00680i)13-s + (−2.69 + 2.59i)14-s + 6.10·15-s + (3.99 + 0.0893i)16-s + (1.52 + 2.63i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.00558i)2-s + (0.470 − 1.75i)3-s + (0.999 + 0.0111i)4-s + (0.224 + 0.837i)5-s + (−0.480 + 1.75i)6-s + (0.717 − 0.696i)7-s + (−0.999 − 0.0167i)8-s + (−1.99 − 1.15i)9-s + (−0.219 − 0.838i)10-s + (−0.337 − 0.0904i)11-s + (0.490 − 1.75i)12-s + (0.00188 + 0.00188i)13-s + (−0.721 + 0.692i)14-s + 1.57·15-s + (0.999 + 0.0223i)16-s + (0.369 + 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.630275 - 0.538274i\)
\(L(\frac12)\) \(\approx\) \(0.630275 - 0.538274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.00789i)T \)
7 \( 1 + (-1.89 + 1.84i)T \)
good3 \( 1 + (-0.814 + 3.04i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.501 - 1.87i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.11 + 0.299i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.00680 - 0.00680i)T + 13iT^{2} \)
17 \( 1 + (-1.52 - 2.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.64 - 1.24i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.27 - 2.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.45 + 1.45i)T + 29iT^{2} \)
31 \( 1 + (-2.60 - 4.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.28 - 8.51i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.02iT - 41T^{2} \)
43 \( 1 + (-7.17 + 7.17i)T - 43iT^{2} \)
47 \( 1 + (-0.796 + 1.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.787 + 0.211i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (7.25 + 1.94i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (9.21 - 2.46i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.70 - 6.35i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 6.77iT - 71T^{2} \)
73 \( 1 + (3.43 - 1.98i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.81 - 4.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.3 + 11.3i)T + 83iT^{2} \)
89 \( 1 + (-3.59 - 2.07i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.390T + 97T^{2} \)
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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

−13.38370818046758936152322169476, −12.31659262768692336956116619589, −11.21473065362111978441851378759, −10.34739985730653320403052328671, −8.666759209794128696775262119051, −7.84637308450292195623594094728, −7.04452813295194113993083887933, −6.16103392350177065948430114375, −2.88626313927546453903524909063, −1.49018620735571660448969639560, 2.63081805891666699767060431361, 4.57975057295458502149132843884, 5.64983474697250140927462926566, 7.897143614500718195905327438420, 8.995298089851989763757381056934, 9.234932578801735542968564259048, 10.55362822811028217596216587424, 11.24444133605222884353282790046, 12.60519852466989140261273525693, 14.38380037190124213008732968554

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