Properties

Label 2-112-16.11-c4-0-12
Degree $2$
Conductor $112$
Sign $-0.796 - 0.604i$
Analytic cond. $11.5774$
Root an. cond. $3.40256$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.78 + 1.29i)2-s + (11.4 + 11.4i)3-s + (12.6 − 9.77i)4-s + (8.99 + 8.99i)5-s + (−57.9 − 28.4i)6-s − 18.5·7-s + (−35.3 + 53.3i)8-s + 179. i·9-s + (−45.6 − 22.4i)10-s + (16.8 − 16.8i)11-s + (256. + 33.0i)12-s + (−139. + 139. i)13-s + (70.1 − 23.8i)14-s + 205. i·15-s + (65.0 − 247. i)16-s + 327.·17-s + ⋯
L(s)  = 1  + (−0.946 + 0.322i)2-s + (1.26 + 1.26i)3-s + (0.791 − 0.610i)4-s + (0.359 + 0.359i)5-s + (−1.60 − 0.791i)6-s − 0.377·7-s + (−0.552 + 0.833i)8-s + 2.21i·9-s + (−0.456 − 0.224i)10-s + (0.139 − 0.139i)11-s + (1.77 + 0.229i)12-s + (−0.828 + 0.828i)13-s + (0.357 − 0.121i)14-s + 0.912i·15-s + (0.254 − 0.967i)16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.796 - 0.604i$
Analytic conductor: \(11.5774\)
Root analytic conductor: \(3.40256\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :2),\ -0.796 - 0.604i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.516300 + 1.53328i\)
\(L(\frac12)\) \(\approx\) \(0.516300 + 1.53328i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.78 - 1.29i)T \)
7 \( 1 + 18.5T \)
good3 \( 1 + (-11.4 - 11.4i)T + 81iT^{2} \)
5 \( 1 + (-8.99 - 8.99i)T + 625iT^{2} \)
11 \( 1 + (-16.8 + 16.8i)T - 1.46e4iT^{2} \)
13 \( 1 + (139. - 139. i)T - 2.85e4iT^{2} \)
17 \( 1 - 327.T + 8.35e4T^{2} \)
19 \( 1 + (-134. - 134. i)T + 1.30e5iT^{2} \)
23 \( 1 + 352.T + 2.79e5T^{2} \)
29 \( 1 + (533. - 533. i)T - 7.07e5iT^{2} \)
31 \( 1 - 1.49e3iT - 9.23e5T^{2} \)
37 \( 1 + (-431. - 431. i)T + 1.87e6iT^{2} \)
41 \( 1 + 2.82e3iT - 2.82e6T^{2} \)
43 \( 1 + (341. - 341. i)T - 3.41e6iT^{2} \)
47 \( 1 + 3.99e3iT - 4.87e6T^{2} \)
53 \( 1 + (-524. - 524. i)T + 7.89e6iT^{2} \)
59 \( 1 + (-25.1 + 25.1i)T - 1.21e7iT^{2} \)
61 \( 1 + (-202. + 202. i)T - 1.38e7iT^{2} \)
67 \( 1 + (-5.55e3 - 5.55e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 4.83e3T + 2.54e7T^{2} \)
73 \( 1 + 4.64e3iT - 2.83e7T^{2} \)
79 \( 1 - 2.21e3iT - 3.89e7T^{2} \)
83 \( 1 + (-8.54e3 - 8.54e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 8.35e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.25e4T + 8.85e7T^{2} \)
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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

−14.11169081134672099313990023609, −12.04945140755255247812886589950, −10.50800250997656536007496297920, −9.944468381964191867181825830279, −9.184042189305949829915586830873, −8.209176335814292400027266749209, −6.99864124554335985592335930287, −5.30743362638740072534057569467, −3.52131998178796575188286845629, −2.19874254998880322585836902173, 0.825923580158743685806764667880, 2.18268114483842811884596984980, 3.30273747006446449654007422147, 6.13386528306694923670228458443, 7.54855250485884729495081784387, 7.930250474543082565137895385507, 9.324481262045473650340717869182, 9.805403754009979118612502315461, 11.65442228945317491021487844266, 12.68636165822596755756077508195

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