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Properties

Label 2-112-112.109-c1-0-6
Degree $2$
Conductor $112$
Sign $0.903 + 0.428i$
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  + (−1.25 − 0.656i)2-s + (1.95 − 0.523i)3-s + (1.13 + 1.64i)4-s + (0.959 + 0.256i)5-s + (−2.78 − 0.625i)6-s + (0.292 + 2.62i)7-s + (−0.348 − 2.80i)8-s + (0.941 − 0.543i)9-s + (−1.03 − 0.951i)10-s + (−0.505 − 1.88i)11-s + (3.08 + 2.61i)12-s + (−2.10 − 2.10i)13-s + (1.35 − 3.48i)14-s + 2.00·15-s + (−1.40 + 3.74i)16-s + (2.83 − 4.91i)17-s + ⋯
L(s)  = 1  + (−0.885 − 0.463i)2-s + (1.12 − 0.302i)3-s + (0.569 + 0.821i)4-s + (0.428 + 0.114i)5-s + (−1.13 − 0.255i)6-s + (0.110 + 0.993i)7-s + (−0.123 − 0.992i)8-s + (0.313 − 0.181i)9-s + (−0.326 − 0.300i)10-s + (−0.152 − 0.569i)11-s + (0.890 + 0.754i)12-s + (−0.583 − 0.583i)13-s + (0.363 − 0.931i)14-s + 0.518·15-s + (−0.351 + 0.936i)16-s + (0.688 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.903 + 0.428i$
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: \((2,\ 112,\ (\ :1/2),\ 0.903 + 0.428i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.979942 - 0.220311i\)
\(L(\frac12)\) \(\approx\) \(0.979942 - 0.220311i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 + 0.656i)T \)
7 \( 1 + (-0.292 - 2.62i)T \)
good3 \( 1 + (-1.95 + 0.523i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.959 - 0.256i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.505 + 1.88i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.10 + 2.10i)T + 13iT^{2} \)
17 \( 1 + (-2.83 + 4.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.165 - 0.616i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.92 - 3.42i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.207 + 0.207i)T + 29iT^{2} \)
31 \( 1 + (3.94 - 6.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.60 - 2.57i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.40iT - 41T^{2} \)
43 \( 1 + (-3.65 + 3.65i)T - 43iT^{2} \)
47 \( 1 + (0.144 + 0.250i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.04 + 7.62i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.60 - 13.4i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.969 - 3.61i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-10.0 + 2.69i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + (0.310 + 0.179i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.84 + 6.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.424 + 0.424i)T + 83iT^{2} \)
89 \( 1 + (15.2 - 8.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−13.55252842204296786552501905477, −12.40495135761884091010438423212, −11.49718581143019340468488745102, −10.03748074890913196530581359136, −9.243472959866402121635461235840, −8.271728623192330963600513851871, −7.48348393658602595090043705815, −5.72557988952444330436847728023, −3.18851252327284185206124206212, −2.20017665802828857809636013800, 2.10553907798920651262428439582, 4.16301043298586113160516880217, 6.04889503291608106717977910372, 7.51565281142020130906060106460, 8.214080630312675007265462609699, 9.594377846686010809417859975539, 9.932432650614330452212730239954, 11.26357004561257900786346486672, 12.90436229381170896518933257327, 14.29077839099426313605761454512

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