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Properties

Label 2-112-28.19-c5-0-7
Degree $2$
Conductor $112$
Sign $0.989 + 0.145i$
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  + (−13.7 − 23.7i)3-s + (1.65 + 0.953i)5-s + (62.6 + 113. i)7-s + (−254. + 440. i)9-s + (237. − 137. i)11-s + 886. i·13-s − 52.2i·15-s + (−746. + 431. i)17-s + (609. − 1.05e3i)19-s + (1.83e3 − 3.04e3i)21-s + (2.95e3 + 1.70e3i)23-s + (−1.56e3 − 2.70e3i)25-s + 7.30e3·27-s + 7.55e3·29-s + (−1.62e3 − 2.82e3i)31-s + ⋯
L(s)  = 1  + (−0.879 − 1.52i)3-s + (0.0295 + 0.0170i)5-s + (0.482 + 0.875i)7-s + (−1.04 + 1.81i)9-s + (0.592 − 0.342i)11-s + 1.45i·13-s − 0.0600i·15-s + (−0.626 + 0.361i)17-s + (0.387 − 0.670i)19-s + (0.909 − 1.50i)21-s + (1.16 + 0.671i)23-s + (−0.499 − 0.865i)25-s + 1.92·27-s + 1.66·29-s + (−0.304 − 0.527i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.989 + 0.145i$
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: \((2,\ 112,\ (\ :5/2),\ 0.989 + 0.145i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.315071408\)
\(L(\frac12)\) \(\approx\) \(1.315071408\)
\(L(\frac{7}{2})\) 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 \( 1 + (-62.6 - 113. i)T \)
good3 \( 1 + (13.7 + 23.7i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-1.65 - 0.953i)T + (1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-237. + 137. i)T + (8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 886. iT - 3.71e5T^{2} \)
17 \( 1 + (746. - 431. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-609. + 1.05e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-2.95e3 - 1.70e3i)T + (3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 7.55e3T + 2.05e7T^{2} \)
31 \( 1 + (1.62e3 + 2.82e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-516. + 895. i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 9.64e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.30e4iT - 1.47e8T^{2} \)
47 \( 1 + (-375. + 650. i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.09e4 - 1.90e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (2.10e4 + 3.64e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.89e4 - 1.09e4i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-4.61e4 + 2.66e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.92e4iT - 1.80e9T^{2} \)
73 \( 1 + (-5.62e4 + 3.24e4i)T + (1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (2.56e4 + 1.48e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.74e4T + 3.93e9T^{2} \)
89 \( 1 + (-6.65e3 - 3.84e3i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.51e5iT - 8.58e9T^{2} \)
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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

−12.50126985659255772867500119607, −11.59814437557963908387489231743, −11.22960239811456951823749938082, −9.222685644856070755941487833713, −8.142246262911929296485023716756, −6.82416747157222908658928971895, −6.16543900857018626387268104430, −4.81043880190885020091545727077, −2.31979868710772600020882869310, −1.12342035663346243961800146109, 0.68545717933090968692636984685, 3.46427334648967526913359717387, 4.60867634977270900234173261660, 5.50164092454384199132329586919, 6.99788712991323676505594655499, 8.612596545696566453279837014198, 9.889147921075188235314701428776, 10.55559126035670139822331831160, 11.33162769860460059513944548556, 12.46300575113670214261653414182

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