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Properties

Label 2-112-112.101-c2-0-25
Degree $2$
Conductor $112$
Sign $0.999 + 0.0217i$
Analytic cond. $3.05177$
Root an. cond. $1.74693$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.96 + 0.352i)2-s + (4.78 − 1.28i)3-s + (3.75 + 1.38i)4-s + (−6.80 − 1.82i)5-s + (9.86 − 0.838i)6-s + (−4.05 + 5.70i)7-s + (6.89 + 4.05i)8-s + (13.4 − 7.75i)9-s + (−12.7 − 5.98i)10-s + (−5.70 + 1.52i)11-s + (19.7 + 1.82i)12-s + (−13.8 − 13.8i)13-s + (−9.99 + 9.79i)14-s − 34.8·15-s + (12.1 + 10.4i)16-s + (3.82 + 2.20i)17-s + ⋯
L(s)  = 1  + (0.984 + 0.176i)2-s + (1.59 − 0.427i)3-s + (0.937 + 0.346i)4-s + (−1.36 − 0.364i)5-s + (1.64 − 0.139i)6-s + (−0.579 + 0.814i)7-s + (0.862 + 0.506i)8-s + (1.49 − 0.861i)9-s + (−1.27 − 0.598i)10-s + (−0.518 + 0.139i)11-s + (1.64 + 0.152i)12-s + (−1.06 − 1.06i)13-s + (−0.714 + 0.699i)14-s − 2.32·15-s + (0.759 + 0.650i)16-s + (0.225 + 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0217i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.999 + 0.0217i$
Analytic conductor: \(3.05177\)
Root analytic conductor: \(1.74693\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1),\ 0.999 + 0.0217i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.91519 - 0.0316954i\)
\(L(\frac12)\) \(\approx\) \(2.91519 - 0.0316954i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.96 - 0.352i)T \)
7 \( 1 + (4.05 - 5.70i)T \)
good3 \( 1 + (-4.78 + 1.28i)T + (7.79 - 4.5i)T^{2} \)
5 \( 1 + (6.80 + 1.82i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (5.70 - 1.52i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (13.8 + 13.8i)T + 169iT^{2} \)
17 \( 1 + (-3.82 - 2.20i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.09 - 11.5i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-28.8 + 16.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-6.91 + 6.91i)T - 841iT^{2} \)
31 \( 1 + (-7.09 - 4.09i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (18.2 - 68.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 14.0T + 1.68e3T^{2} \)
43 \( 1 + (25.7 + 25.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (-50.5 + 29.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-22.8 + 6.11i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (16.3 + 61.0i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-11.5 + 42.9i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-6.02 - 22.4i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 68.1iT - 5.04e3T^{2} \)
73 \( 1 + (-5.88 + 10.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-1.88 - 3.27i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (76.0 + 76.0i)T + 6.88e3iT^{2} \)
89 \( 1 + (-54.1 - 93.7i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 41.0iT - 9.40e3T^{2} \)
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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

−13.24944214656805801076686659078, −12.57900337406172609515570618051, −11.95716077925997343650280519823, −10.17403678635130903048461772748, −8.529498898712756154593772248785, −7.943781489924957374420801795471, −6.94409961480490481654490451225, −5.01914231805261105569783957529, −3.49659899945437724770108427852, −2.63539244706271585545620632813, 2.74531305759373959054044169530, 3.69242590337816893929802806155, 4.59597538718479696024868713538, 7.15040457418206107789804008680, 7.52454681786038656289184238924, 9.131020312032154013678915155614, 10.32129081357396543000496585106, 11.37477564443380578316826290296, 12.61431480027079657502607322631, 13.62497073205156550428045829231

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