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Properties

Label 2-112-112.101-c2-0-23
Degree $2$
Conductor $112$
Sign $0.457 + 0.889i$
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  + (1.21 − 1.58i)2-s + (2.57 − 0.689i)3-s + (−1.04 − 3.86i)4-s + (5.82 + 1.56i)5-s + (2.03 − 4.92i)6-s + (1.48 + 6.84i)7-s + (−7.40 − 3.03i)8-s + (−1.65 + 0.956i)9-s + (9.56 − 7.35i)10-s + (−7.58 + 2.03i)11-s + (−5.34 − 9.20i)12-s + (−13.1 − 13.1i)13-s + (12.6 + 5.96i)14-s + 16.0·15-s + (−13.8 + 8.06i)16-s + (3.45 + 1.99i)17-s + ⋯
L(s)  = 1  + (0.607 − 0.794i)2-s + (0.857 − 0.229i)3-s + (−0.261 − 0.965i)4-s + (1.16 + 0.312i)5-s + (0.338 − 0.820i)6-s + (0.211 + 0.977i)7-s + (−0.925 − 0.379i)8-s + (−0.184 + 0.106i)9-s + (0.956 − 0.735i)10-s + (−0.689 + 0.184i)11-s + (−0.445 − 0.767i)12-s + (−1.00 − 1.00i)13-s + (0.904 + 0.426i)14-s + 1.07·15-s + (−0.863 + 0.504i)16-s + (0.203 + 0.117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.457 + 0.889i$
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: \((2,\ 112,\ (\ :1),\ 0.457 + 0.889i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.06315 - 1.25843i\)
\(L(\frac12)\) \(\approx\) \(2.06315 - 1.25843i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 + 1.58i)T \)
7 \( 1 + (-1.48 - 6.84i)T \)
good3 \( 1 + (-2.57 + 0.689i)T + (7.79 - 4.5i)T^{2} \)
5 \( 1 + (-5.82 - 1.56i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (7.58 - 2.03i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (13.1 + 13.1i)T + 169iT^{2} \)
17 \( 1 + (-3.45 - 1.99i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (0.217 - 0.810i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-4.14 + 2.39i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-3.85 + 3.85i)T - 841iT^{2} \)
31 \( 1 + (-17.1 - 9.92i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-15.7 + 58.8i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 34.6T + 1.68e3T^{2} \)
43 \( 1 + (-54.0 - 54.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (55.3 - 31.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-47.3 + 12.6i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (4.49 + 16.7i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-30.3 + 113. i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-19.9 - 74.4i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 117. iT - 5.04e3T^{2} \)
73 \( 1 + (3.92 - 6.78i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-26.6 - 46.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (48.3 + 48.3i)T + 6.88e3iT^{2} \)
89 \( 1 + (76.8 + 133. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 127. iT - 9.40e3T^{2} \)
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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

−13.08408473973694570136164151854, −12.55066673282486355372251137245, −11.10301503328901442464113480197, −10.00582878653696162749170346980, −9.213514978943787822970232334609, −7.88739074814927229016372887700, −5.98397658932304925589223730207, −5.12467821638692632900546273690, −2.84218963311975825880672907737, −2.23289638688786819242922961157, 2.61402076107405835189870280158, 4.25837339868290182783116085227, 5.49943677187450133202541883489, 6.88296428225818478082045345778, 8.036917841477135048287872366475, 9.160113815924382551933306007595, 10.03988052558867745252128090619, 11.72972768592494080457354636722, 13.11493758847739740411416681236, 13.80995539144069864909197821536

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