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Properties

Label 2-112-112.27-c1-0-4
Degree $2$
Conductor $112$
Sign $-0.315 - 0.948i$
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 + (−1.73 + 1.73i)3-s + 2i·4-s + (1.73 − 1.73i)5-s − 3.46·6-s + (−2 + 1.73i)7-s + (−2 + 2i)8-s − 2.99i·9-s + 3.46·10-s + (3 − 3i)11-s + (−3.46 − 3.46i)12-s + (1.73 + 1.73i)13-s + (−3.73 − 0.267i)14-s + 5.99i·15-s − 4·16-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.999 + 0.999i)3-s + i·4-s + (0.774 − 0.774i)5-s − 1.41·6-s + (−0.755 + 0.654i)7-s + (−0.707 + 0.707i)8-s − 0.999i·9-s + 1.09·10-s + (0.904 − 0.904i)11-s + (−0.999 − 0.999i)12-s + (0.480 + 0.480i)13-s + (−0.997 − 0.0716i)14-s + 1.54i·15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.315 - 0.948i$
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: \((2,\ 112,\ (\ :1/2),\ -0.315 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.680949 + 0.944045i\)
\(L(\frac12)\) \(\approx\) \(0.680949 + 0.944045i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good3 \( 1 + (1.73 - 1.73i)T - 3iT^{2} \)
5 \( 1 + (-1.73 + 1.73i)T - 5iT^{2} \)
11 \( 1 + (-3 + 3i)T - 11iT^{2} \)
13 \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + (-5.19 + 5.19i)T - 19iT^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (1 - i)T - 29iT^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + (1 - i)T - 43iT^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + (-3 - 3i)T + 53iT^{2} \)
59 \( 1 + (-8.66 - 8.66i)T + 59iT^{2} \)
61 \( 1 + (5.19 + 5.19i)T + 61iT^{2} \)
67 \( 1 + (7 + 7i)T + 67iT^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + (1.73 - 1.73i)T - 83iT^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−13.85784703037706909770905382852, −13.04793151320571880107117797183, −11.89266318232035543858969068563, −11.09602482688579785538965267093, −9.328393493192492217290321254576, −8.975360543397950453492636898554, −6.73696702817197666026046957078, −5.70005969049126727731688382625, −5.08279188106306037424145639994, −3.55331439759956367832127603927, 1.51683417946449306790458795206, 3.48998008248667682799886741427, 5.46588591591552423050078067013, 6.46478331865508771838043689610, 7.09857301973895263508066601972, 9.628841156639319640668093914213, 10.38403295975674916887185207614, 11.44279943089379227248756714241, 12.35787669372916781342984631960, 13.14941108790057849941748651185

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