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Properties

Label 2-112-28.19-c5-0-12
Degree $2$
Conductor $112$
Sign $0.403 + 0.914i$
Analytic cond. $17.9629$
Root an. cond. $4.23827$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.33 − 5.76i)3-s + (0.990 + 0.571i)5-s + (129. − 2.46i)7-s + (99.3 − 172. i)9-s + (−510. + 294. i)11-s + 480. i·13-s − 7.61i·15-s + (1.80e3 − 1.04e3i)17-s + (972. − 1.68e3i)19-s + (−445. − 739. i)21-s + (−729. − 421. i)23-s + (−1.56e3 − 2.70e3i)25-s − 2.94e3·27-s + 8.11e3·29-s + (−3.79e3 − 6.57e3i)31-s + ⋯
L(s)  = 1  + (−0.213 − 0.370i)3-s + (0.0177 + 0.0102i)5-s + (0.999 − 0.0190i)7-s + (0.408 − 0.707i)9-s + (−1.27 + 0.734i)11-s + 0.788i·13-s − 0.00874i·15-s + (1.51 − 0.873i)17-s + (0.617 − 1.07i)19-s + (−0.220 − 0.365i)21-s + (−0.287 − 0.166i)23-s + (−0.499 − 0.865i)25-s − 0.776·27-s + 1.79·29-s + (−0.709 − 1.22i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.403 + 0.914i$
Analytic conductor: \(17.9629\)
Root analytic conductor: \(4.23827\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :5/2),\ 0.403 + 0.914i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.808948333\)
\(L(\frac12)\) \(\approx\) \(1.808948333\)
\(L(\frac{7}{2})\) 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 \( 1 + (-129. + 2.46i)T \)
good3 \( 1 + (3.33 + 5.76i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-0.990 - 0.571i)T + (1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (510. - 294. i)T + (8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 480. iT - 3.71e5T^{2} \)
17 \( 1 + (-1.80e3 + 1.04e3i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-972. + 1.68e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (729. + 421. i)T + (3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 8.11e3T + 2.05e7T^{2} \)
31 \( 1 + (3.79e3 + 6.57e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-4.79e3 + 8.29e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 9.68e3iT - 1.15e8T^{2} \)
43 \( 1 + 4.32e3iT - 1.47e8T^{2} \)
47 \( 1 + (3.12e3 - 5.41e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-3.69e3 - 6.39e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.81e4 - 3.15e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-273. - 157. i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.39e4 - 1.38e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.14e4iT - 1.80e9T^{2} \)
73 \( 1 + (6.00e4 - 3.46e4i)T + (1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-8.04e4 - 4.64e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 2.43e4T + 3.93e9T^{2} \)
89 \( 1 + (-1.10e4 - 6.40e3i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 9.37e4iT - 8.58e9T^{2} \)
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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

−12.28357419693937277588999853189, −11.70103213217379095510934105093, −10.37501832573367980390800300577, −9.353613731325634877855982978410, −7.85728429094650940647966155104, −7.11373469801287568422677728071, −5.54594295438506597283419145532, −4.38411312463621066321733604517, −2.40446194573021934818092658982, −0.78499043359460808974721863682, 1.39854312121743040601634073757, 3.27672318000133519819730620205, 4.98799403920849600858442075180, 5.69504649104032744227513432805, 7.83854574369979735509742263612, 8.123918685269502382310769886494, 10.11840530693980818529338607070, 10.55441574510027435776425167151, 11.73609537924674771040441393764, 12.89450354628036265626639383617

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