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Properties

Label 2-112-112.93-c1-0-9
Degree $2$
Conductor $112$
Sign $0.968 - 0.247i$
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  + (1.01 + 0.988i)2-s + (0.430 − 1.60i)3-s + (0.0448 + 1.99i)4-s + (−0.522 − 1.94i)5-s + (2.02 − 1.20i)6-s + (1.89 + 1.85i)7-s + (−1.93 + 2.06i)8-s + (0.196 + 0.113i)9-s + (1.39 − 2.48i)10-s + (−6.09 − 1.63i)11-s + (3.23 + 0.789i)12-s + (1.13 + 1.13i)13-s + (0.0819 + 3.74i)14-s − 3.35·15-s + (−3.99 + 0.179i)16-s + (−0.960 − 1.66i)17-s + ⋯
L(s)  = 1  + (0.714 + 0.699i)2-s + (0.248 − 0.928i)3-s + (0.0224 + 0.999i)4-s + (−0.233 − 0.871i)5-s + (0.827 − 0.490i)6-s + (0.714 + 0.699i)7-s + (−0.682 + 0.730i)8-s + (0.0655 + 0.0378i)9-s + (0.442 − 0.786i)10-s + (−1.83 − 0.492i)11-s + (0.933 + 0.227i)12-s + (0.314 + 0.314i)13-s + (0.0219 + 0.999i)14-s − 0.867·15-s + (−0.998 + 0.0448i)16-s + (−0.233 − 0.403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.968 - 0.247i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.968 - 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47450 + 0.185135i\)
\(L(\frac12)\) \(\approx\) \(1.47450 + 0.185135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.01 - 0.988i)T \)
7 \( 1 + (-1.89 - 1.85i)T \)
good3 \( 1 + (-0.430 + 1.60i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.522 + 1.94i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (6.09 + 1.63i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.13 - 1.13i)T + 13iT^{2} \)
17 \( 1 + (0.960 + 1.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.09 - 1.63i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.924 - 0.533i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.08 - 5.08i)T + 29iT^{2} \)
31 \( 1 + (0.198 + 0.343i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0877 + 0.327i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.26iT - 41T^{2} \)
43 \( 1 + (-1.75 + 1.75i)T - 43iT^{2} \)
47 \( 1 + (1.08 - 1.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.415 + 0.111i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-8.67 - 2.32i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-10.1 + 2.72i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.10 - 7.87i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.78iT - 71T^{2} \)
73 \( 1 + (2.05 - 1.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.10 + 8.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.17 - 1.17i)T + 83iT^{2} \)
89 \( 1 + (11.0 + 6.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.21T + 97T^{2} \)
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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

−13.49298138512998819955449360865, −12.81387367032106462670821242215, −12.17721787007972471874926807041, −10.83119605516369078446944358711, −8.517033164519237628130499124591, −8.310077780187395550574585289610, −7.06944170955387723988713013902, −5.59963190664780951947391070878, −4.61706924696377925337007580752, −2.42994317770750623460224168276, 2.65916244160702260606668778904, 4.07383698765311021034459331051, 5.00254639791816647231824108481, 6.74989147891538397584188175172, 8.221900300028250841878357827560, 9.961436702459392242391807878666, 10.60469732215519782945103124497, 11.08792250479548091957036997475, 12.71433999969454158509450411194, 13.53363593374520206191028341966

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