Properties

Label 2-112-28.11-c4-0-13
Degree $2$
Conductor $112$
Sign $-0.838 + 0.544i$
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  + (6.40 + 3.69i)3-s + (−10.1 − 17.6i)5-s + (−48.6 + 5.63i)7-s + (−13.1 − 22.7i)9-s + (−47.8 − 27.6i)11-s − 237.·13-s − 150. i·15-s + (−131. + 227. i)17-s + (388. − 224. i)19-s + (−332. − 143. i)21-s + (79.1 − 45.6i)23-s + (105. − 182. i)25-s − 793. i·27-s − 1.36e3·29-s + (−471. − 272. i)31-s + ⋯
L(s)  = 1  + (0.711 + 0.410i)3-s + (−0.407 − 0.705i)5-s + (−0.993 + 0.115i)7-s + (−0.162 − 0.280i)9-s + (−0.395 − 0.228i)11-s − 1.40·13-s − 0.669i·15-s + (−0.455 + 0.788i)17-s + (1.07 − 0.622i)19-s + (−0.754 − 0.326i)21-s + (0.149 − 0.0863i)23-s + (0.168 − 0.292i)25-s − 1.08i·27-s − 1.62·29-s + (−0.491 − 0.283i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.838 + 0.544i$
Analytic conductor: \(11.5774\)
Root analytic conductor: \(3.40256\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :2),\ -0.838 + 0.544i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.148975 - 0.503331i\)
\(L(\frac12)\) \(\approx\) \(0.148975 - 0.503331i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (48.6 - 5.63i)T \)
good3 \( 1 + (-6.40 - 3.69i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (10.1 + 17.6i)T + (-312.5 + 541. i)T^{2} \)
11 \( 1 + (47.8 + 27.6i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + 237.T + 2.85e4T^{2} \)
17 \( 1 + (131. - 227. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-388. + 224. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-79.1 + 45.6i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + 1.36e3T + 7.07e5T^{2} \)
31 \( 1 + (471. + 272. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-421. - 730. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 1.72e3T + 2.82e6T^{2} \)
43 \( 1 - 1.11e3iT - 3.41e6T^{2} \)
47 \( 1 + (-1.05e3 + 609. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-1.97e3 + 3.41e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-2.00e3 - 1.15e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-1.60e3 - 2.77e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-3.80e3 - 2.19e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 4.52e3iT - 2.54e7T^{2} \)
73 \( 1 + (371. - 643. i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (8.78e3 - 5.07e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.04e4iT - 4.74e7T^{2} \)
89 \( 1 + (2.87e3 + 4.98e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 1.25e4T + 8.85e7T^{2} \)
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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

−12.60834444323595019885658115802, −11.58896379236296747762065352329, −10.00280975756573017162409597460, −9.280354190635792272681155946962, −8.323444560204136520687167022210, −7.01452194337730832930063544001, −5.40938205768751835036584520089, −3.96004101167463639729461570600, −2.71206388016176823729481752133, −0.19504738132560090166448117460, 2.40708162057824058157867348601, 3.42897456479291443308293090742, 5.34666054833255663999642630536, 7.19867914945866450377479661197, 7.46573843830966146941369274392, 9.106644145090530600280324281679, 10.04528920843940868345664199807, 11.26906734692368294448092714369, 12.45939032890130005967051515449, 13.38830451385746317188570109871

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