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Properties

Label 2-112-28.3-c3-0-9
Degree $2$
Conductor $112$
Sign $0.0742 + 0.997i$
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  + (3.53 − 6.11i)3-s + (1.05 − 0.611i)5-s + (16.5 − 8.40i)7-s + (−11.4 − 19.8i)9-s + (3.96 + 2.28i)11-s − 41.0i·13-s − 8.64i·15-s + (−103. − 59.7i)17-s + (30.0 + 52.1i)19-s + (6.85 − 130. i)21-s + (−7.23 + 4.17i)23-s + (−61.7 + 106. i)25-s + 29.0·27-s + 164.·29-s + (143. − 249. i)31-s + ⋯
L(s)  = 1  + (0.679 − 1.17i)3-s + (0.0948 − 0.0547i)5-s + (0.891 − 0.453i)7-s + (−0.423 − 0.733i)9-s + (0.108 + 0.0627i)11-s − 0.876i·13-s − 0.148i·15-s + (−1.47 − 0.852i)17-s + (0.363 + 0.629i)19-s + (0.0712 − 1.35i)21-s + (−0.0655 + 0.0378i)23-s + (−0.494 + 0.855i)25-s + 0.207·27-s + 1.05·29-s + (0.833 − 1.44i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0742 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0742 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.0742 + 0.997i$
Analytic conductor: \(6.60821\)
Root analytic conductor: \(2.57064\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :3/2),\ 0.0742 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.52165 - 1.41262i\)
\(L(\frac12)\) \(\approx\) \(1.52165 - 1.41262i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-16.5 + 8.40i)T \)
good3 \( 1 + (-3.53 + 6.11i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-1.05 + 0.611i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-3.96 - 2.28i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 41.0iT - 2.19e3T^{2} \)
17 \( 1 + (103. + 59.7i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-30.0 - 52.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (7.23 - 4.17i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 164.T + 2.43e4T^{2} \)
31 \( 1 + (-143. + 249. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-74.3 - 128. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 358. iT - 6.89e4T^{2} \)
43 \( 1 - 360. iT - 7.95e4T^{2} \)
47 \( 1 + (-112. - 195. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (342. - 592. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-42.5 + 73.6i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (181. - 104. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-361. - 208. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 982. iT - 3.57e5T^{2} \)
73 \( 1 + (-295. - 170. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-66.0 + 38.1i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 523.T + 5.71e5T^{2} \)
89 \( 1 + (841. - 486. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 676. iT - 9.12e5T^{2} \)
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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

−13.21663647619111136826026823260, −11.97413595355420823171427525054, −10.97491030165431942490184936665, −9.524761096741203912481515394096, −8.151972394445048697972119750219, −7.62826461797634484906073269266, −6.35373796362067836939390798326, −4.63863813678811383062416246697, −2.67292974588638560053783593835, −1.20038085593285846228435690301, 2.27606039398354135455343091874, 4.00909589020085908742519289306, 4.95030485419732023669836998772, 6.66960941955780813576229890183, 8.466898402216145557512239816467, 8.940498072384205485499978205127, 10.18594942041268437766633484809, 11.11159815226072847181603780538, 12.21633208546743383391155258116, 13.82794829871946542654039553831

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