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Properties

Label 2-112-112.19-c1-0-2
Degree $2$
Conductor $112$
Sign $0.225 - 0.974i$
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  + (−1.29 + 0.573i)2-s + (−0.388 + 1.44i)3-s + (1.34 − 1.48i)4-s + (2.81 − 0.755i)5-s + (−0.328 − 2.09i)6-s + (−1.47 + 2.19i)7-s + (−0.885 + 2.68i)8-s + (0.649 + 0.375i)9-s + (−3.21 + 2.59i)10-s + (0.0165 − 0.0619i)11-s + (1.62 + 2.52i)12-s + (−3.62 + 3.62i)13-s + (0.651 − 3.68i)14-s + 4.37i·15-s + (−0.396 − 3.98i)16-s + (3.26 − 1.88i)17-s + ⋯
L(s)  = 1  + (−0.914 + 0.405i)2-s + (−0.224 + 0.836i)3-s + (0.671 − 0.741i)4-s + (1.26 − 0.337i)5-s + (−0.134 − 0.855i)6-s + (−0.558 + 0.829i)7-s + (−0.312 + 0.949i)8-s + (0.216 + 0.125i)9-s + (−1.01 + 0.819i)10-s + (0.00500 − 0.0186i)11-s + (0.469 + 0.727i)12-s + (−1.00 + 1.00i)13-s + (0.174 − 0.984i)14-s + 1.12i·15-s + (−0.0990 − 0.995i)16-s + (0.790 − 0.456i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.225 - 0.974i$
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: \((2,\ 112,\ (\ :1/2),\ 0.225 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.596777 + 0.474535i\)
\(L(\frac12)\) \(\approx\) \(0.596777 + 0.474535i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 - 0.573i)T \)
7 \( 1 + (1.47 - 2.19i)T \)
good3 \( 1 + (0.388 - 1.44i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-2.81 + 0.755i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.0165 + 0.0619i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.62 - 3.62i)T - 13iT^{2} \)
17 \( 1 + (-3.26 + 1.88i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.13 + 0.572i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.80 + 6.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.74 + 4.74i)T + 29iT^{2} \)
31 \( 1 + (0.329 + 0.570i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.08 + 4.05i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.67T + 41T^{2} \)
43 \( 1 + (2.54 + 2.54i)T + 43iT^{2} \)
47 \( 1 + (-2.62 + 4.55i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.1 + 2.72i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.43 - 0.652i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.84 - 6.89i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-5.98 - 1.60i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 1.08T + 71T^{2} \)
73 \( 1 + (0.0232 + 0.0402i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (15.2 + 8.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.07 + 5.07i)T + 83iT^{2} \)
89 \( 1 + (4.47 - 7.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.85iT - 97T^{2} \)
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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

−14.16210083803036003328007055954, −12.75981342379428690402336358356, −11.50304871089677819193595266965, −10.15633698863017766846047055409, −9.580908597509974895770861180199, −8.982808447924691220349731228278, −7.20324209024652381940144239348, −5.90214720710000953036243939878, −4.99116487502974878628110074465, −2.28155763328023644291768988080, 1.39953530974462614901713681202, 3.15190895584211554286265761598, 5.81642531279984782880976988870, 7.01973176220009585658715708344, 7.69102678677671418188001405531, 9.613395060022002324801177188517, 9.958571607369797145662790294729, 11.11777071057778580946455774013, 12.61494025517879331795343052446, 12.97697272517088337752606077260

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