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Properties

Label 2-112-28.19-c5-0-17
Degree $2$
Conductor $112$
Sign $0.110 + 0.993i$
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  + (6.90 + 11.9i)3-s + (−4.24 − 2.45i)5-s + (−30.4 − 126. i)7-s + (26.0 − 45.1i)9-s + (−465. + 268. i)11-s − 1.07e3i·13-s − 67.7i·15-s + (−1.11e3 + 641. i)17-s + (1.00e3 − 1.73e3i)19-s + (1.29e3 − 1.23e3i)21-s + (2.50e3 + 1.44e3i)23-s + (−1.55e3 − 2.68e3i)25-s + 4.07e3·27-s − 2.30e3·29-s + (−980. − 1.69e3i)31-s + ⋯
L(s)  = 1  + (0.443 + 0.767i)3-s + (−0.0759 − 0.0438i)5-s + (−0.234 − 0.972i)7-s + (0.107 − 0.185i)9-s + (−1.16 + 0.670i)11-s − 1.75i·13-s − 0.0777i·15-s + (−0.931 + 0.538i)17-s + (0.636 − 1.10i)19-s + (0.641 − 0.610i)21-s + (0.987 + 0.570i)23-s + (−0.496 − 0.859i)25-s + 1.07·27-s − 0.507·29-s + (−0.183 − 0.317i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.110 + 0.993i$
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: \((2,\ 112,\ (\ :5/2),\ 0.110 + 0.993i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.359685768\)
\(L(\frac12)\) \(\approx\) \(1.359685768\)
\(L(\frac{7}{2})\) 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 \( 1 + (30.4 + 126. i)T \)
good3 \( 1 + (-6.90 - 11.9i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (4.24 + 2.45i)T + (1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (465. - 268. i)T + (8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 1.07e3iT - 3.71e5T^{2} \)
17 \( 1 + (1.11e3 - 641. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.00e3 + 1.73e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-2.50e3 - 1.44e3i)T + (3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 2.30e3T + 2.05e7T^{2} \)
31 \( 1 + (980. + 1.69e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.28e3 + 9.15e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 9.16e3iT - 1.15e8T^{2} \)
43 \( 1 + 579. iT - 1.47e8T^{2} \)
47 \( 1 + (4.67e3 - 8.10e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.71e4 + 2.96e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (7.14 + 12.3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.60e4 - 1.50e4i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (5.17e3 - 2.98e3i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 4.69e4iT - 1.80e9T^{2} \)
73 \( 1 + (5.61e4 - 3.24e4i)T + (1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-4.91e4 - 2.83e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 3.57e4T + 3.93e9T^{2} \)
89 \( 1 + (-7.62e4 - 4.40e4i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 3.38e4iT - 8.58e9T^{2} \)
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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

−12.83746798943427592622543251175, −11.01793933833877052737834365757, −10.28270590127361230365005130793, −9.428872136217877278527681191064, −8.056326112415386286290559849152, −7.05161073550835788911927772671, −5.27827196356911989112824496559, −4.06729677237173821672432035643, −2.82541451834427993811630867539, −0.46680429480210106207951939860, 1.76893715778536974572660756998, 2.93966016400440793005354345320, 4.90335013973853724322933072910, 6.32475374353814485112907260373, 7.47600167855194493864167880151, 8.541960036237551522640323372873, 9.489698386310793809553366345572, 11.02847765765951472170337701495, 11.96643845275886096823185199855, 13.10237848792239248023201467233

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