Properties

Label 2-112-28.3-c3-0-5
Degree $2$
Conductor $112$
Sign $0.667 - 0.744i$
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  + (−0.5 + 0.866i)3-s + (−4.5 + 2.59i)5-s + (14 − 12.1i)7-s + (13 + 22.5i)9-s + (22.5 + 12.9i)11-s + 69.2i·13-s − 5.19i·15-s + (31.5 + 18.1i)17-s + (8.5 + 14.7i)19-s + (3.5 + 18.1i)21-s + (121.5 − 70.1i)23-s + (−49 + 84.8i)25-s − 53·27-s + 90·29-s + (−8.5 + 14.7i)31-s + ⋯
L(s)  = 1  + (−0.0962 + 0.166i)3-s + (−0.402 + 0.232i)5-s + (0.755 − 0.654i)7-s + (0.481 + 0.833i)9-s + (0.616 + 0.356i)11-s + 1.47i·13-s − 0.0894i·15-s + (0.449 + 0.259i)17-s + (0.102 + 0.177i)19-s + (0.0363 + 0.188i)21-s + (1.10 − 0.635i)23-s + (−0.392 + 0.678i)25-s − 0.377·27-s + 0.576·29-s + (−0.0492 + 0.0852i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.667 - 0.744i$
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: \((2,\ 112,\ (\ :3/2),\ 0.667 - 0.744i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.45567 + 0.650479i\)
\(L(\frac12)\) \(\approx\) \(1.45567 + 0.650479i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (4.5 - 2.59i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-22.5 - 12.9i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 69.2iT - 2.19e3T^{2} \)
17 \( 1 + (-31.5 - 18.1i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-8.5 - 14.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-121.5 + 70.1i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 90T + 2.43e4T^{2} \)
31 \( 1 + (8.5 - 14.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (99.5 + 172. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 187. iT - 6.89e4T^{2} \)
43 \( 1 - 252. iT - 7.95e4T^{2} \)
47 \( 1 + (283.5 + 491. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-166.5 + 288. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (400.5 - 693. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (310.5 - 179. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (187.5 + 108. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 488. iT - 3.57e5T^{2} \)
73 \( 1 + (-349.5 - 201. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-1.03e3 + 596. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 468T + 5.71e5T^{2} \)
89 \( 1 + (166.5 - 96.1i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.39e3iT - 9.12e5T^{2} \)
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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

−13.43889098003630781318289833507, −12.05115306550780090487314236709, −11.17753660438961887839456111288, −10.28903657111439428336160040551, −8.984234520350558678377662665603, −7.65217838418125466251495668060, −6.81177956970681326298237333720, −4.92355969118957217379179870001, −3.94522498795801371093276684657, −1.67673324021216051478958280464, 1.03297301311748921299127165765, 3.24835159982206457382141625806, 4.88325203345011471344931012578, 6.15203693084191986172668840553, 7.60619864054561974000385032259, 8.593257525884054257091566411157, 9.733009354133742288989243740934, 11.11812697405302105441483570519, 12.03794186359490671223511060122, 12.76179935428352140991660546308

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