↑

Properties

Label 2-112-7.3-c6-0-0
Degree $2$
Conductor $112$
Sign $0.308 - 0.951i$
Analytic cond. $25.7660$
Root an. cond. $5.07602$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−21.8 − 12.6i)3-s + (−10.7 + 6.20i)5-s + (−195. − 281. i)7-s + (−45.0 − 78.0i)9-s + (−774. + 1.34e3i)11-s − 2.77e3i·13-s + 313.·15-s + (−109. − 63.2i)17-s + (1.14e3 − 662. i)19-s + (723. + 8.63e3i)21-s + (−7.11e3 − 1.23e4i)23-s + (−7.73e3 + 1.33e4i)25-s + 2.07e4i·27-s − 7.47e3·29-s + (4.92e4 + 2.84e4i)31-s + ⋯
L(s)  = 1  + (−0.810 − 0.468i)3-s + (−0.0859 + 0.0496i)5-s + (−0.570 − 0.821i)7-s + (−0.0618 − 0.107i)9-s + (−0.582 + 1.00i)11-s − 1.26i·13-s + 0.0929·15-s + (−0.0223 − 0.0128i)17-s + (0.167 − 0.0966i)19-s + (0.0781 + 0.932i)21-s + (−0.585 − 1.01i)23-s + (−0.495 + 0.857i)25-s + 1.05i·27-s − 0.306·29-s + (1.65 + 0.954i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.308 - 0.951i$
Analytic conductor: \(25.7660\)
Root analytic conductor: \(5.07602\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :3),\ 0.308 - 0.951i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.4560599590\)
\(L(\frac12)\) \(\approx\) \(0.4560599590\)
\(L(4)\) 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 \( 1 + (195. + 281. i)T \)
good3 \( 1 + (21.8 + 12.6i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (10.7 - 6.20i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (774. - 1.34e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 2.77e3iT - 4.82e6T^{2} \)
17 \( 1 + (109. + 63.2i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-1.14e3 + 662. i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (7.11e3 + 1.23e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 7.47e3T + 5.94e8T^{2} \)
31 \( 1 + (-4.92e4 - 2.84e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-4.50e4 - 7.79e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 3.57e4iT - 4.75e9T^{2} \)
43 \( 1 - 7.94e4T + 6.32e9T^{2} \)
47 \( 1 + (1.26e5 - 7.29e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-8.50e4 + 1.47e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-1.87e5 - 1.08e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.72e5 - 9.95e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-7.22e4 + 1.25e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 4.07e5T + 1.28e11T^{2} \)
73 \( 1 + (1.61e5 + 9.33e4i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-4.19e4 - 7.26e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 1.62e5iT - 3.26e11T^{2} \)
89 \( 1 + (4.39e5 - 2.53e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 5.09e5iT - 8.32e11T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.70182865232860659729018041434, −11.73709987071011935177312843191, −10.53284744729778368553734623050, −9.808829832200012595902609075473, −8.057049022639702049571494019108, −7.02741241611973186059883974304, −6.03032494059021233980109997828, −4.68590955199281651505704089242, −3.02262571026228653924856368506, −0.987403361946188564968488208638, 0.20547731795407960249842420826, 2.42758795507810600613286356218, 4.11445180402672517615172502236, 5.54373569419943395709023172267, 6.23555838076202023817795564758, 7.947393297671431828712817908801, 9.164938772504746427777809571905, 10.19109800050179041716930511066, 11.40211407268793884609638649407, 11.89885546534585736361844313302

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