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Properties

Label 2-112-16.3-c2-0-9
Degree $2$
Conductor $112$
Sign $-0.294 - 0.955i$
Analytic cond. $3.05177$
Root an. cond. $1.74693$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.99 + 0.0471i)2-s + (−2.60 + 2.60i)3-s + (3.99 + 0.188i)4-s + (−5.26 + 5.26i)5-s + (−5.33 + 5.08i)6-s − 2.64·7-s + (7.98 + 0.565i)8-s − 4.57i·9-s + (−10.7 + 10.2i)10-s + (−0.0342 − 0.0342i)11-s + (−10.8 + 9.91i)12-s + (5.31 + 5.31i)13-s + (−5.29 − 0.124i)14-s − 27.4i·15-s + (15.9 + 1.50i)16-s + 27.5·17-s + ⋯
L(s)  = 1  + (0.999 + 0.0235i)2-s + (−0.868 + 0.868i)3-s + (0.998 + 0.0471i)4-s + (−1.05 + 1.05i)5-s + (−0.888 + 0.847i)6-s − 0.377·7-s + (0.997 + 0.0706i)8-s − 0.507i·9-s + (−1.07 + 1.02i)10-s + (−0.00311 − 0.00311i)11-s + (−0.908 + 0.826i)12-s + (0.408 + 0.408i)13-s + (−0.377 − 0.00890i)14-s − 1.83i·15-s + (0.995 + 0.0941i)16-s + 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.294 - 0.955i$
Analytic conductor: \(3.05177\)
Root analytic conductor: \(1.74693\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1),\ -0.294 - 0.955i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.920680 + 1.24648i\)
\(L(\frac12)\) \(\approx\) \(0.920680 + 1.24648i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 - 0.0471i)T \)
7 \( 1 + 2.64T \)
good3 \( 1 + (2.60 - 2.60i)T - 9iT^{2} \)
5 \( 1 + (5.26 - 5.26i)T - 25iT^{2} \)
11 \( 1 + (0.0342 + 0.0342i)T + 121iT^{2} \)
13 \( 1 + (-5.31 - 5.31i)T + 169iT^{2} \)
17 \( 1 - 27.5T + 289T^{2} \)
19 \( 1 + (1.06 - 1.06i)T - 361iT^{2} \)
23 \( 1 + 8.59T + 529T^{2} \)
29 \( 1 + (-35.6 - 35.6i)T + 841iT^{2} \)
31 \( 1 + 49.7iT - 961T^{2} \)
37 \( 1 + (28.9 - 28.9i)T - 1.36e3iT^{2} \)
41 \( 1 + 47.8iT - 1.68e3T^{2} \)
43 \( 1 + (8.62 + 8.62i)T + 1.84e3iT^{2} \)
47 \( 1 - 79.5iT - 2.20e3T^{2} \)
53 \( 1 + (-26.1 + 26.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (-28.4 - 28.4i)T + 3.48e3iT^{2} \)
61 \( 1 + (39.9 + 39.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (-59.3 + 59.3i)T - 4.48e3iT^{2} \)
71 \( 1 + 76.2T + 5.04e3T^{2} \)
73 \( 1 - 103. iT - 5.32e3T^{2} \)
79 \( 1 - 45.1iT - 6.24e3T^{2} \)
83 \( 1 + (-55.5 + 55.5i)T - 6.88e3iT^{2} \)
89 \( 1 + 51.0iT - 7.92e3T^{2} \)
97 \( 1 + 27.5T + 9.40e3T^{2} \)
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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

−13.90647433888466414362738601178, −12.33667086584135268763767789154, −11.62997542607776815754908795935, −10.80060210642619346015877661867, −10.05790383353639716684302520386, −7.86095830292316083944297879634, −6.71211459624465801568228539403, −5.58501835125917499414172522957, −4.21000067499882213679412126997, −3.23566304611137927756320205147, 0.994277964015690272733621527246, 3.57326764297711064358981213734, 5.03228867664878463201531957852, 6.07603580248205554601055300301, 7.29625414067463776233318027884, 8.307236958505736414154235491405, 10.29081510551740704601080557768, 11.68671114819655284670226980666, 12.16178570118483287815988604114, 12.74413916114297413948308499248

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