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Properties

Label 2-112-28.27-c1-0-2
Degree $2$
Conductor $112$
Sign $0.188 + 0.981i$
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  − 2·3-s − 3.46i·5-s + (2 − 1.73i)7-s + 9-s − 3.46i·11-s + 3.46i·13-s + 6.92i·15-s − 2·19-s + (−4 + 3.46i)21-s + 3.46i·23-s − 6.99·25-s + 4·27-s + 6·29-s + 8·31-s + 6.92i·33-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.54i·5-s + (0.755 − 0.654i)7-s + 0.333·9-s − 1.04i·11-s + 0.960i·13-s + 1.78i·15-s − 0.458·19-s + (−0.872 + 0.755i)21-s + 0.722i·23-s − 1.39·25-s + 0.769·27-s + 1.11·29-s + 1.43·31-s + 1.20i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.188 + 0.981i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.188 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.568112 - 0.469204i\)
\(L(\frac12)\) \(\approx\) \(0.568112 - 0.469204i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good3 \( 1 + 2T + 3T^{2} \)
5 \( 1 + 3.46iT - 5T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−13.34745702644632746708570537502, −12.08541445430052409448284292638, −11.53608786648513702801776631862, −10.48361246250146110961431272083, −8.989465811714025912703719229052, −8.088033745491439757133644511878, −6.40412089618040462057025792788, −5.21407291662131394303443555357, −4.35574014289993715868955404535, −1.03873125222266531521343601974, 2.64170589847875004777786885404, 4.77705997943313852717858901537, 6.03477501177299682547156576526, 6.95137221767744892846555518300, 8.275593580169004230850885191372, 10.16572090900810806046058816110, 10.71746733338320226481159020141, 11.71600842780928516730164992035, 12.43629648787716540674983378319, 14.05149851150536601912731627835

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