Properties

Label 2-112-7.4-c7-0-24
Degree $2$
Conductor $112$
Sign $-0.995 - 0.0970i$
Analytic cond. $34.9871$
Root an. cond. $5.91499$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (29.4 − 50.9i)3-s + (−212. − 367. i)5-s + (30.7 − 906. i)7-s + (−635. − 1.10e3i)9-s + (3.94e3 − 6.83e3i)11-s + 6.71e3·13-s − 2.49e4·15-s + (3.53e3 − 6.12e3i)17-s + (−1.31e4 − 2.27e4i)19-s + (−4.52e4 − 2.82e4i)21-s + (6.03e3 + 1.04e4i)23-s + (−5.09e4 + 8.82e4i)25-s + 5.38e4·27-s + 3.06e3·29-s + (−5.92e4 + 1.02e5i)31-s + ⋯
L(s)  = 1  + (0.628 − 1.08i)3-s + (−0.758 − 1.31i)5-s + (0.0338 − 0.999i)7-s + (−0.290 − 0.503i)9-s + (0.893 − 1.54i)11-s + 0.848·13-s − 1.90·15-s + (0.174 − 0.302i)17-s + (−0.438 − 0.760i)19-s + (−1.06 − 0.665i)21-s + (0.103 + 0.179i)23-s + (−0.652 + 1.12i)25-s + 0.526·27-s + 0.0233·29-s + (−0.357 + 0.618i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0970i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.995 - 0.0970i$
Analytic conductor: \(34.9871\)
Root analytic conductor: \(5.91499\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :7/2),\ -0.995 - 0.0970i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.267183087\)
\(L(\frac12)\) \(\approx\) \(2.267183087\)
\(L(\frac{9}{2})\) 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 \( 1 + (-30.7 + 906. i)T \)
good3 \( 1 + (-29.4 + 50.9i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (212. + 367. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (-3.94e3 + 6.83e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 - 6.71e3T + 6.27e7T^{2} \)
17 \( 1 + (-3.53e3 + 6.12e3i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (1.31e4 + 2.27e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-6.03e3 - 1.04e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 3.06e3T + 1.72e10T^{2} \)
31 \( 1 + (5.92e4 - 1.02e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-2.29e5 - 3.98e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 - 3.16e5T + 1.94e11T^{2} \)
43 \( 1 - 3.16e4T + 2.71e11T^{2} \)
47 \( 1 + (-4.03e5 - 6.98e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (2.39e5 - 4.15e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-3.37e5 + 5.84e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-2.83e5 - 4.90e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-5.92e5 + 1.02e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 4.61e6T + 9.09e12T^{2} \)
73 \( 1 + (1.52e6 - 2.63e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (3.45e6 + 5.97e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 9.01e6T + 2.71e13T^{2} \)
89 \( 1 + (-3.50e6 - 6.07e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 8.60e6T + 8.07e13T^{2} \)
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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

−11.85204933878369653569809362965, −10.94298435331818736969077589644, −9.028601113524045145143877705213, −8.401772269957105519986694163516, −7.51289288117623184292945541352, −6.28028768096994226734287621772, −4.51345708171829566470826947184, −3.32906670482530602455260132359, −1.21707929115132173591256957364, −0.73454415030337869311993039489, 2.19472175431953741923489999809, 3.52229771399934831796976343184, 4.28500277298336204006802927049, 6.13235558697110103356457587253, 7.37621860062251855269795182283, 8.670071585118868809731097290439, 9.635578667311363615351270966108, 10.56980078800333808614008717012, 11.60331315677249121121416143610, 12.59770933197380644766001953830

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