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Properties

Label 2-112-112.53-c1-0-11
Degree $2$
Conductor $112$
Sign $0.279 + 0.960i$
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  + (0.559 − 1.29i)2-s + (0.224 + 0.839i)3-s + (−1.37 − 1.45i)4-s + (0.847 − 3.16i)5-s + (1.21 + 0.177i)6-s + (−0.654 + 2.56i)7-s + (−2.65 + 0.968i)8-s + (1.94 − 1.12i)9-s + (−3.63 − 2.87i)10-s + (2.87 − 0.769i)11-s + (0.911 − 1.47i)12-s + (−3.63 + 3.63i)13-s + (2.96 + 2.28i)14-s + 2.84·15-s + (−0.229 + 3.99i)16-s + (−1.81 + 3.14i)17-s + ⋯
L(s)  = 1  + (0.395 − 0.918i)2-s + (0.129 + 0.484i)3-s + (−0.686 − 0.727i)4-s + (0.379 − 1.41i)5-s + (0.496 + 0.0726i)6-s + (−0.247 + 0.968i)7-s + (−0.939 + 0.342i)8-s + (0.648 − 0.374i)9-s + (−1.14 − 0.908i)10-s + (0.865 − 0.231i)11-s + (0.263 − 0.426i)12-s + (−1.00 + 1.00i)13-s + (0.791 + 0.610i)14-s + 0.734·15-s + (−0.0574 + 0.998i)16-s + (−0.441 + 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.279 + 0.960i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.279 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00741 - 0.755831i\)
\(L(\frac12)\) \(\approx\) \(1.00741 - 0.755831i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.559 + 1.29i)T \)
7 \( 1 + (0.654 - 2.56i)T \)
good3 \( 1 + (-0.224 - 0.839i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.847 + 3.16i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.87 + 0.769i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.63 - 3.63i)T - 13iT^{2} \)
17 \( 1 + (1.81 - 3.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.60 - 0.429i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.33 - 3.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.10 + 5.10i)T - 29iT^{2} \)
31 \( 1 + (1.00 - 1.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.49 + 5.57i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.71iT - 41T^{2} \)
43 \( 1 + (2.91 + 2.91i)T + 43iT^{2} \)
47 \( 1 + (5.06 + 8.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.68 + 0.986i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.64 - 0.977i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.54 + 1.75i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.57 + 5.88i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.55iT - 71T^{2} \)
73 \( 1 + (0.989 + 0.571i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.120 - 0.209i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.459 - 0.459i)T - 83iT^{2} \)
89 \( 1 + (-3.76 + 2.17i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.80T + 97T^{2} \)
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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

−13.17411375399885183175577984355, −12.22350306214123826279588869459, −11.79395076454031676219094490308, −9.956318934586339450207488502347, −9.364310685597800733808352106879, −8.647625804148090726834729972565, −6.21530042535009233145114151201, −4.94717277177938885117712792300, −3.96004501770319414576577037629, −1.82674729320211907682559746413, 2.97254012049944524773591715810, 4.58005225605456781388041011019, 6.40988201763444123296502387692, 7.06842215851006500728995862973, 7.81497808526472623520597566645, 9.682053968887918908792258404911, 10.49470258886773283927995378247, 12.07582909019758829353244557632, 13.17523142915879432602334389652, 14.04880755635619671654366206837

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