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Properties

Label 2-112-112.109-c1-0-12
Degree $2$
Conductor $112$
Sign $-0.0588 + 0.998i$
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  + (−0.246 − 1.39i)2-s + (2.51 − 0.672i)3-s + (−1.87 + 0.687i)4-s + (−2.91 − 0.780i)5-s + (−1.55 − 3.33i)6-s + (1.41 − 2.23i)7-s + (1.42 + 2.44i)8-s + (3.25 − 1.87i)9-s + (−0.367 + 4.24i)10-s + (0.838 + 3.12i)11-s + (−4.25 + 2.98i)12-s + (2.52 + 2.52i)13-s + (−3.45 − 1.42i)14-s − 7.83·15-s + (3.05 − 2.58i)16-s + (0.201 − 0.348i)17-s + ⋯
L(s)  = 1  + (−0.174 − 0.984i)2-s + (1.44 − 0.388i)3-s + (−0.939 + 0.343i)4-s + (−1.30 − 0.348i)5-s + (−0.635 − 1.35i)6-s + (0.536 − 0.843i)7-s + (0.502 + 0.864i)8-s + (1.08 − 0.626i)9-s + (−0.116 + 1.34i)10-s + (0.252 + 0.943i)11-s + (−1.22 + 0.862i)12-s + (0.699 + 0.699i)13-s + (−0.924 − 0.381i)14-s − 2.02·15-s + (0.763 − 0.645i)16-s + (0.0487 − 0.0844i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0588 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0588 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.0588 + 0.998i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ -0.0588 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.805625 - 0.854555i\)
\(L(\frac12)\) \(\approx\) \(0.805625 - 0.854555i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.246 + 1.39i)T \)
7 \( 1 + (-1.41 + 2.23i)T \)
good3 \( 1 + (-2.51 + 0.672i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (2.91 + 0.780i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.838 - 3.12i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.52 - 2.52i)T + 13iT^{2} \)
17 \( 1 + (-0.201 + 0.348i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.373 - 1.39i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (7.89 - 4.55i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.47 - 1.47i)T + 29iT^{2} \)
31 \( 1 + (-2.12 + 3.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.94 - 0.520i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.96iT - 41T^{2} \)
43 \( 1 + (0.997 - 0.997i)T - 43iT^{2} \)
47 \( 1 + (2.09 + 3.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.488 - 1.82i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.636 + 2.37i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.685 - 2.55i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (11.6 - 3.12i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.451iT - 71T^{2} \)
73 \( 1 + (9.40 + 5.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.31 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.742 - 0.742i)T + 83iT^{2} \)
89 \( 1 + (-11.1 + 6.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−13.43124260043410321330383641603, −12.22348076128423833748157606686, −11.49684726696262501560430071011, −10.10620553379742253707051433570, −8.950337971699468948279588200079, −8.027499287537131901547243263559, −7.44143417926766444532989753807, −4.30013961292181821528611961365, −3.69881246923569502285332472889, −1.76290398642672750531741360422, 3.22682757635517175108955319512, 4.37915757208225022133451869484, 6.17439571096651179976354461170, 7.969556100869516092126570272590, 8.183884472588649161348145215176, 9.071225047078923840196551875444, 10.50325129664698054642235719483, 11.86616884420796543902486155168, 13.33085868318304179893776651210, 14.37057255861568957032930249397

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