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Properties

Label 2-112-7.2-c3-0-9
Degree $2$
Conductor $112$
Sign $-0.605 + 0.795i$
Analytic cond. $6.60821$
Root an. cond. $2.57064$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.5 − 4.33i)3-s + (4.5 − 7.79i)5-s + (14 + 12.1i)7-s + (0.999 − 1.73i)9-s + (−28.5 − 49.3i)11-s − 70·13-s − 45.0·15-s + (−25.5 − 44.1i)17-s + (2.5 − 4.33i)19-s + (17.5 − 90.9i)21-s + (34.5 − 59.7i)23-s + (22 + 38.1i)25-s − 144.·27-s + 114·29-s + (11.5 + 19.9i)31-s + ⋯
L(s)  = 1  + (−0.481 − 0.833i)3-s + (0.402 − 0.697i)5-s + (0.755 + 0.654i)7-s + (0.0370 − 0.0641i)9-s + (−0.781 − 1.35i)11-s − 1.49·13-s − 0.774·15-s + (−0.363 − 0.630i)17-s + (0.0301 − 0.0522i)19-s + (0.181 − 0.944i)21-s + (0.312 − 0.541i)23-s + (0.175 + 0.304i)25-s − 1.03·27-s + 0.729·29-s + (0.0666 + 0.115i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(6.60821\)
Root analytic conductor: \(2.57064\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :3/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.536093 - 1.08142i\)
\(L(\frac12)\) \(\approx\) \(0.536093 - 1.08142i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14 - 12.1i)T \)
good3 \( 1 + (2.5 + 4.33i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-4.5 + 7.79i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (28.5 + 49.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 70T + 2.19e3T^{2} \)
17 \( 1 + (25.5 + 44.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-34.5 + 59.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 114T + 2.43e4T^{2} \)
31 \( 1 + (-11.5 - 19.9i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-126.5 + 219. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 42T + 6.89e4T^{2} \)
43 \( 1 - 124T + 7.95e4T^{2} \)
47 \( 1 + (-100.5 + 174. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-196.5 - 340. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-109.5 - 189. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-354.5 + 614. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-209.5 - 362. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 96T + 3.57e5T^{2} \)
73 \( 1 + (-156.5 - 271. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-230.5 + 399. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 588T + 5.71e5T^{2} \)
89 \( 1 + (-508.5 + 880. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.83e3T + 9.12e5T^{2} \)
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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

−12.66837681099206710461923172594, −11.94157378300821387265262199356, −10.92312515150809899999584688326, −9.408072128830228046476294858753, −8.371835607442038289036460901721, −7.20352261558888564560603235313, −5.78360694535227747061701620313, −4.93853429110145066101634325214, −2.43984266907745830284815425869, −0.68833255702630976640643116535, 2.27991706446225172238720019559, 4.40566887810134080486787917437, 5.16332039021654588844110979129, 6.92071386332276326703817062407, 7.86698863812020588520640571484, 9.822062218638266939600391947917, 10.24787097418557539767231598488, 11.12617753905737023704994500914, 12.38589878609169320169845936982, 13.60782332685305149632238179424

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