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Properties

Label 2-112-28.19-c5-0-14
Degree $2$
Conductor $112$
Sign $0.955 - 0.294i$
Analytic cond. $17.9629$
Root an. cond. $4.23827$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.77 + 8.26i)3-s + (88.0 + 50.8i)5-s + (12.6 − 129. i)7-s + (75.9 − 131. i)9-s + (496. − 286. i)11-s − 25.1i·13-s + 971. i·15-s + (−1.12e3 + 651. i)17-s + (425. − 736. i)19-s + (1.12e3 − 511. i)21-s + (−1.90e3 − 1.09e3i)23-s + (3.61e3 + 6.25e3i)25-s + 3.76e3·27-s + 7.13e3·29-s + (−1.63e3 − 2.82e3i)31-s + ⋯
L(s)  = 1  + (0.306 + 0.530i)3-s + (1.57 + 0.909i)5-s + (0.0975 − 0.995i)7-s + (0.312 − 0.541i)9-s + (1.23 − 0.714i)11-s − 0.0412i·13-s + 1.11i·15-s + (−0.947 + 0.547i)17-s + (0.270 − 0.468i)19-s + (0.557 − 0.253i)21-s + (−0.750 − 0.433i)23-s + (1.15 + 2.00i)25-s + 0.995·27-s + 1.57·29-s + (−0.305 − 0.528i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.955 - 0.294i$
Analytic conductor: \(17.9629\)
Root analytic conductor: \(4.23827\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :5/2),\ 0.955 - 0.294i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.943895679\)
\(L(\frac12)\) \(\approx\) \(2.943895679\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-12.6 + 129. i)T \)
good3 \( 1 + (-4.77 - 8.26i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-88.0 - 50.8i)T + (1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-496. + 286. i)T + (8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 25.1iT - 3.71e5T^{2} \)
17 \( 1 + (1.12e3 - 651. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-425. + 736. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.90e3 + 1.09e3i)T + (3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 7.13e3T + 2.05e7T^{2} \)
31 \( 1 + (1.63e3 + 2.82e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (6.76e3 - 1.17e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 169. iT - 1.15e8T^{2} \)
43 \( 1 - 1.74e4iT - 1.47e8T^{2} \)
47 \( 1 + (325. - 563. i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-4.90e3 - 8.49e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (2.07e4 + 3.58e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.00e4 - 5.82e3i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.65e4 - 1.53e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.27e4iT - 1.80e9T^{2} \)
73 \( 1 + (1.80e4 - 1.03e4i)T + (1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.11e4 - 6.43e3i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.20e5T + 3.93e9T^{2} \)
89 \( 1 + (8.64e4 + 4.98e4i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 4.67e4iT - 8.58e9T^{2} \)
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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

−13.06378443768384856423704199169, −11.40411693689319566616087230686, −10.35894594588869482361511468724, −9.754401878010174241930932602096, −8.697546361673974000962081607153, −6.76760584384211362344986808729, −6.26136178411592354114125126266, −4.36109145381792165057503933080, −3.06037470063180009154285120813, −1.34975869737327388180596091771, 1.52080765717446512997275924342, 2.23667847152223807780217032988, 4.68624855400508258663468346596, 5.79406535529085324421104262544, 6.94406001638923325916148021518, 8.610732597622702224767121904269, 9.233605568327657873655774930832, 10.24632896686805724268112795441, 12.02261263083122158694710767641, 12.62153984763329771966654389404

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