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Properties

Label 2-112-112.19-c1-0-11
Degree $2$
Conductor $112$
Sign $0.520 + 0.853i$
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  + (1.12 − 0.863i)2-s + (0.0530 − 0.198i)3-s + (0.509 − 1.93i)4-s + (−1.82 + 0.487i)5-s + (−0.111 − 0.267i)6-s + (1.84 − 1.89i)7-s + (−1.09 − 2.60i)8-s + (2.56 + 1.47i)9-s + (−1.61 + 2.11i)10-s + (−1.09 + 4.07i)11-s + (−0.356 − 0.203i)12-s + (−2.63 + 2.63i)13-s + (0.437 − 3.71i)14-s + 0.386i·15-s + (−3.48 − 1.96i)16-s + (−5.54 + 3.20i)17-s + ⋯
L(s)  = 1  + (0.792 − 0.610i)2-s + (0.0306 − 0.114i)3-s + (0.254 − 0.967i)4-s + (−0.814 + 0.218i)5-s + (−0.0455 − 0.109i)6-s + (0.698 − 0.715i)7-s + (−0.388 − 0.921i)8-s + (0.853 + 0.492i)9-s + (−0.511 + 0.669i)10-s + (−0.328 + 1.22i)11-s + (−0.102 − 0.0587i)12-s + (−0.729 + 0.729i)13-s + (0.117 − 0.993i)14-s + 0.0997i·15-s + (−0.870 − 0.492i)16-s + (−1.34 + 0.776i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.520 + 0.853i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.520 + 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26002 - 0.707223i\)
\(L(\frac12)\) \(\approx\) \(1.26002 - 0.707223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.12 + 0.863i)T \)
7 \( 1 + (-1.84 + 1.89i)T \)
good3 \( 1 + (-0.0530 + 0.198i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (1.82 - 0.487i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.09 - 4.07i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.63 - 2.63i)T - 13iT^{2} \)
17 \( 1 + (5.54 - 3.20i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.83 + 1.29i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.51 + 4.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0380 - 0.0380i)T + 29iT^{2} \)
31 \( 1 + (2.32 + 4.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.349 + 1.30i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 + (3.36 + 3.36i)T + 43iT^{2} \)
47 \( 1 + (1.66 - 2.88i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.28 + 1.14i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.75 + 0.739i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.44 + 5.39i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-7.68 - 2.05i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + (1.44 + 2.50i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.46 - 3.15i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.36 + 5.36i)T + 83iT^{2} \)
89 \( 1 + (0.890 - 1.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.40iT - 97T^{2} \)
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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

−13.30980617424538446831461958020, −12.49471699104549058161953971436, −11.39920044483345062011525220527, −10.64872395558631517203570815086, −9.539681384190605695102618118569, −7.60692516253863901243721675824, −6.91309677045019617757501092812, −4.76531308224216899254042096785, −4.18403445974245974691265224742, −2.05134853734184419780379530202, 3.12048562488134542247559895899, 4.58846511821549714061640895171, 5.63401278626575310731307788332, 7.22672805012713971212345456291, 8.103467150338102257625282879953, 9.222129193356392619708358978783, 11.13270326246337693014841603292, 11.86253712968622592959569089519, 12.81735526537097257427838068448, 13.85188508094204150808743927566

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