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Properties

Label 2-112-7.2-c9-0-13
Degree $2$
Conductor $112$
Sign $-0.975 + 0.218i$
Analytic cond. $57.6840$
Root an. cond. $7.59499$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (131. + 227. i)3-s + (53.0 − 91.8i)5-s + (992. + 6.27e3i)7-s + (−2.46e4 + 4.27e4i)9-s + (4.53e4 + 7.85e4i)11-s − 6.65e4·13-s + 2.78e4·15-s + (−2.12e5 − 3.68e5i)17-s + (2.70e5 − 4.68e5i)19-s + (−1.29e6 + 1.05e6i)21-s + (−5.87e5 + 1.01e6i)23-s + (9.70e5 + 1.68e6i)25-s − 7.81e6·27-s + 6.66e6·29-s + (8.35e5 + 1.44e6i)31-s + ⋯
L(s)  = 1  + (0.936 + 1.62i)3-s + (0.0379 − 0.0657i)5-s + (0.156 + 0.987i)7-s + (−1.25 + 2.17i)9-s + (0.934 + 1.61i)11-s − 0.646·13-s + 0.142·15-s + (−0.618 − 1.07i)17-s + (0.475 − 0.824i)19-s + (−1.45 + 1.17i)21-s + (−0.437 + 0.758i)23-s + (0.497 + 0.861i)25-s − 2.82·27-s + 1.75·29-s + (0.162 + 0.281i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.975 + 0.218i$
Analytic conductor: \(57.6840\)
Root analytic conductor: \(7.59499\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :9/2),\ -0.975 + 0.218i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.623752391\)
\(L(\frac12)\) \(\approx\) \(2.623752391\)
\(L(\frac{11}{2})\) 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 \( 1 + (-992. - 6.27e3i)T \)
good3 \( 1 + (-131. - 227. i)T + (-9.84e3 + 1.70e4i)T^{2} \)
5 \( 1 + (-53.0 + 91.8i)T + (-9.76e5 - 1.69e6i)T^{2} \)
11 \( 1 + (-4.53e4 - 7.85e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + 6.65e4T + 1.06e10T^{2} \)
17 \( 1 + (2.12e5 + 3.68e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-2.70e5 + 4.68e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (5.87e5 - 1.01e6i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 - 6.66e6T + 1.45e13T^{2} \)
31 \( 1 + (-8.35e5 - 1.44e6i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + (-2.81e6 + 4.86e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 + 1.30e7T + 3.27e14T^{2} \)
43 \( 1 + 1.16e7T + 5.02e14T^{2} \)
47 \( 1 + (-1.35e6 + 2.35e6i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (-1.58e7 - 2.74e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (1.64e7 + 2.84e7i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (-8.57e7 + 1.48e8i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (2.26e7 + 3.92e7i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 - 2.03e8T + 4.58e16T^{2} \)
73 \( 1 + (3.63e7 + 6.29e7i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (-6.81e7 + 1.18e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + 5.10e8T + 1.86e17T^{2} \)
89 \( 1 + (1.52e8 - 2.64e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 + 7.58e8T + 7.60e17T^{2} \)
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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

−12.22319547787024109315566163651, −11.26171515069090092404771113411, −9.770801307898684284205854324814, −9.467708839454301980013695149759, −8.533760009508097410060831347748, −7.08673248751273639957001391944, −5.08049078944686078167872514863, −4.55702061338384597593682447024, −3.08510538530223176958932304674, −2.09012841552664537500374336821, 0.58610998537211277961640202215, 1.39866064904950480261906036560, 2.78024742839554718423132102264, 3.93752361016860938190304054916, 6.22144658881345040431790164760, 6.83570479506861873911002412380, 8.175900004704342696688879183229, 8.557812627098329687209652093917, 10.19385147887435405770880931820, 11.58894018978761380715811197011

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