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Properties

Label 2-112-112.27-c1-0-6
Degree $2$
Conductor $112$
Sign $0.999 + 0.00510i$
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 + i)2-s − 2i·4-s + (2.44 − 2.44i)5-s + (−1 − 2.44i)7-s + (2 + 2i)8-s + 3i·9-s + 4.89i·10-s + (1 − i)11-s + (2.44 + 2.44i)13-s + (3.44 + 1.44i)14-s − 4·16-s − 4.89i·17-s + (−3 − 3i)18-s + (−4.89 + 4.89i)19-s + (−4.89 − 4.89i)20-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − i·4-s + (1.09 − 1.09i)5-s + (−0.377 − 0.925i)7-s + (0.707 + 0.707i)8-s + i·9-s + 1.54i·10-s + (0.301 − 0.301i)11-s + (0.679 + 0.679i)13-s + (0.921 + 0.387i)14-s − 16-s − 1.18i·17-s + (−0.707 − 0.707i)18-s + (−1.12 + 1.12i)19-s + (−1.09 − 1.09i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.999 + 0.00510i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.999 + 0.00510i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849818 - 0.00216805i\)
\(L(\frac12)\) \(\approx\) \(0.849818 - 0.00216805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
7 \( 1 + (1 + 2.44i)T \)
good3 \( 1 - 3iT^{2} \)
5 \( 1 + (-2.44 + 2.44i)T - 5iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 + (-2.44 - 2.44i)T + 13iT^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 + (4.89 - 4.89i)T - 19iT^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (3 - 3i)T - 29iT^{2} \)
31 \( 1 + 4.89T + 31T^{2} \)
37 \( 1 + (-5 - 5i)T + 37iT^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 + (-4.89 - 4.89i)T + 59iT^{2} \)
61 \( 1 + (2.44 + 2.44i)T + 61iT^{2} \)
67 \( 1 + (-5 - 5i)T + 67iT^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 9.79T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−13.63885620049263382910389163956, −13.08003031205074378865508171434, −11.19012775524523305971273191730, −10.16610139543642053796701342919, −9.259888402126371369821014914997, −8.351261718427381332149867646789, −6.98771897017817465245209420384, −5.79975670349792459293200198069, −4.63629604979633582067592256342, −1.53789424338446233516979701597, 2.22439406821665087791809461871, 3.51127533700140026182552425123, 6.00918592405223275502832665131, 6.87116364204424014624184509803, 8.679610662984185781541189889030, 9.436737434407069834313449213110, 10.44710081256548129672504128737, 11.28000819759667251586514387054, 12.62408756927549227960372761916, 13.24073753485138955750101158787

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