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Properties

Label 2-112-16.5-c1-0-0
Degree $2$
Conductor $112$
Sign $0.755 - 0.654i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.163 − 1.40i)2-s + (−2.05 + 2.05i)3-s + (−1.94 − 0.459i)4-s + (2.72 + 2.72i)5-s + (2.55 + 3.22i)6-s + i·7-s + (−0.964 + 2.65i)8-s − 5.44i·9-s + (4.27 − 3.38i)10-s + (0.919 + 0.919i)11-s + (4.94 − 3.05i)12-s + (−1.12 + 1.12i)13-s + (1.40 + 0.163i)14-s − 11.2·15-s + (3.57 + 1.78i)16-s − 1.50·17-s + ⋯
L(s)  = 1  + (0.115 − 0.993i)2-s + (−1.18 + 1.18i)3-s + (−0.973 − 0.229i)4-s + (1.21 + 1.21i)5-s + (1.04 + 1.31i)6-s + 0.377i·7-s + (−0.340 + 0.940i)8-s − 1.81i·9-s + (1.35 − 1.07i)10-s + (0.277 + 0.277i)11-s + (1.42 − 0.881i)12-s + (−0.312 + 0.312i)13-s + (0.375 + 0.0437i)14-s − 2.89·15-s + (0.894 + 0.447i)16-s − 0.365·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.755 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759951 + 0.283508i\)
\(L(\frac12)\) \(\approx\) \(0.759951 + 0.283508i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.163 + 1.40i)T \)
7 \( 1 - iT \)
good3 \( 1 + (2.05 - 2.05i)T - 3iT^{2} \)
5 \( 1 + (-2.72 - 2.72i)T + 5iT^{2} \)
11 \( 1 + (-0.919 - 0.919i)T + 11iT^{2} \)
13 \( 1 + (1.12 - 1.12i)T - 13iT^{2} \)
17 \( 1 + 1.50T + 17T^{2} \)
19 \( 1 + (-1.46 + 1.46i)T - 19iT^{2} \)
23 \( 1 + 4.77iT - 23T^{2} \)
29 \( 1 + (-4.10 + 4.10i)T - 29iT^{2} \)
31 \( 1 - 4.10T + 31T^{2} \)
37 \( 1 + (1.65 + 1.65i)T + 37iT^{2} \)
41 \( 1 + 7.45iT - 41T^{2} \)
43 \( 1 + (-5.68 - 5.68i)T + 43iT^{2} \)
47 \( 1 - 3.59T + 47T^{2} \)
53 \( 1 + (0.675 + 0.675i)T + 53iT^{2} \)
59 \( 1 + (-1.13 - 1.13i)T + 59iT^{2} \)
61 \( 1 + (3.21 - 3.21i)T - 61iT^{2} \)
67 \( 1 + (1.52 - 1.52i)T - 67iT^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 - 14.4iT - 73T^{2} \)
79 \( 1 + 1.77T + 79T^{2} \)
83 \( 1 + (7.16 - 7.16i)T - 83iT^{2} \)
89 \( 1 - 8.45iT - 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−13.79237474938703122419226806270, −12.34003895145822980578437762171, −11.38198307931393366711642810372, −10.57683544953828863597613543988, −9.953463813554158145007915304482, −9.151412065694982782918116939611, −6.52097907176943316191726888807, −5.55970946445130325928794471890, −4.34902518198527671741289795438, −2.61137695634340569888499110978, 1.14886678936980541741637656931, 4.87129617476996567624581053944, 5.71118424692006088296210487260, 6.54819930820390365337583100099, 7.74948856138931217127771191727, 8.993394619574307528475281929911, 10.21112358550121999218834719243, 11.89900429417935613256997175993, 12.76362104836096410655862597227, 13.41689759080422966464607955816

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