Properties

Label 2-112-16.11-c4-0-41
Degree $2$
Conductor $112$
Sign $-0.954 - 0.297i$
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  + (−1.61 − 3.66i)2-s + (6.54 + 6.54i)3-s + (−10.7 + 11.8i)4-s + (−10.9 − 10.9i)5-s + (13.3 − 34.5i)6-s − 18.5·7-s + (60.6 + 20.4i)8-s + 4.71i·9-s + (−22.3 + 57.5i)10-s + (−4.98 + 4.98i)11-s + (−147. + 6.68i)12-s + (−55.7 + 55.7i)13-s + (29.8 + 67.7i)14-s − 142. i·15-s + (−23.0 − 254. i)16-s − 492.·17-s + ⋯
L(s)  = 1  + (−0.403 − 0.915i)2-s + (0.727 + 0.727i)3-s + (−0.674 + 0.738i)4-s + (−0.436 − 0.436i)5-s + (0.372 − 0.959i)6-s − 0.377·7-s + (0.947 + 0.319i)8-s + 0.0582i·9-s + (−0.223 + 0.575i)10-s + (−0.0411 + 0.0411i)11-s + (−1.02 + 0.0464i)12-s + (−0.330 + 0.330i)13-s + (0.152 + 0.345i)14-s − 0.634i·15-s + (−0.0901 − 0.995i)16-s − 1.70·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.297i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.954 - 0.297i$
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: \((2,\ 112,\ (\ :2),\ -0.954 - 0.297i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0282386 + 0.185318i\)
\(L(\frac12)\) \(\approx\) \(0.0282386 + 0.185318i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.61 + 3.66i)T \)
7 \( 1 + 18.5T \)
good3 \( 1 + (-6.54 - 6.54i)T + 81iT^{2} \)
5 \( 1 + (10.9 + 10.9i)T + 625iT^{2} \)
11 \( 1 + (4.98 - 4.98i)T - 1.46e4iT^{2} \)
13 \( 1 + (55.7 - 55.7i)T - 2.85e4iT^{2} \)
17 \( 1 + 492.T + 8.35e4T^{2} \)
19 \( 1 + (337. + 337. i)T + 1.30e5iT^{2} \)
23 \( 1 + 290.T + 2.79e5T^{2} \)
29 \( 1 + (1.13e3 - 1.13e3i)T - 7.07e5iT^{2} \)
31 \( 1 - 998. iT - 9.23e5T^{2} \)
37 \( 1 + (319. + 319. i)T + 1.87e6iT^{2} \)
41 \( 1 + 433. iT - 2.82e6T^{2} \)
43 \( 1 + (-499. + 499. i)T - 3.41e6iT^{2} \)
47 \( 1 + 1.76e3iT - 4.87e6T^{2} \)
53 \( 1 + (-3.08e3 - 3.08e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-1.17e3 + 1.17e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-3.81e3 + 3.81e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (722. + 722. i)T + 2.01e7iT^{2} \)
71 \( 1 + 6.98e3T + 2.54e7T^{2} \)
73 \( 1 - 2.56e3iT - 2.83e7T^{2} \)
79 \( 1 - 5.33e3iT - 3.89e7T^{2} \)
83 \( 1 + (7.40e3 + 7.40e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.26e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.21e3T + 8.85e7T^{2} \)
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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

−12.31197193201469665108350732492, −11.13442607497294353628281155337, −10.15999473575069283314488393855, −8.909209507294028091704056651803, −8.736544262680256133163630884412, −7.00693128644314222862125757908, −4.65342680085221120719082704197, −3.77625723989760092201530679518, −2.33928612956340808665343112845, −0.081093710884857513675270491802, 2.14988234930288330040067932472, 4.12580305323421096375144557279, 5.96803628818960693717963317696, 7.10525925573782575945864091274, 7.895199676510640972174543451648, 8.821680667552872908477106435871, 10.05748610794739142361189069932, 11.26332948507242610945126404712, 12.98652175434354928862516736586, 13.47138758847048831607912855306

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