Properties

Label 2-112-7.6-c4-0-3
Degree $2$
Conductor $112$
Sign $-0.142 - 0.989i$
Analytic cond. $11.5774$
Root an. cond. $3.40256$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.92i·3-s − 20.7i·5-s + (7 + 48.4i)7-s + 33.0·9-s − 18·11-s + 131. i·13-s + 144·15-s + 415. i·17-s + 90.0i·19-s + (−336 + 48.4i)21-s − 738·23-s + 193·25-s + 789. i·27-s − 846·29-s + 1.16e3i·31-s + ⋯
L(s)  = 1  + 0.769i·3-s − 0.831i·5-s + (0.142 + 0.989i)7-s + 0.407·9-s − 0.148·11-s + 0.778i·13-s + 0.640·15-s + 1.43i·17-s + 0.249i·19-s + (−0.761 + 0.109i)21-s − 1.39·23-s + 0.308·25-s + 1.08i·27-s − 1.00·29-s + 1.21i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.142 - 0.989i$
Analytic conductor: \(11.5774\)
Root analytic conductor: \(3.40256\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :2),\ -0.142 - 0.989i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.03287 + 1.19266i\)
\(L(\frac12)\) \(\approx\) \(1.03287 + 1.19266i\)
\(L(3)\) 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 \( 1 + (-7 - 48.4i)T \)
good3 \( 1 - 6.92iT - 81T^{2} \)
5 \( 1 + 20.7iT - 625T^{2} \)
11 \( 1 + 18T + 1.46e4T^{2} \)
13 \( 1 - 131. iT - 2.85e4T^{2} \)
17 \( 1 - 415. iT - 8.35e4T^{2} \)
19 \( 1 - 90.0iT - 1.30e5T^{2} \)
23 \( 1 + 738T + 2.79e5T^{2} \)
29 \( 1 + 846T + 7.07e5T^{2} \)
31 \( 1 - 1.16e3iT - 9.23e5T^{2} \)
37 \( 1 - 2.38e3T + 1.87e6T^{2} \)
41 \( 1 - 1.70e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.51e3T + 3.41e6T^{2} \)
47 \( 1 + 3.40e3iT - 4.87e6T^{2} \)
53 \( 1 + 270T + 7.89e6T^{2} \)
59 \( 1 + 3.13e3iT - 1.21e7T^{2} \)
61 \( 1 + 6.49e3iT - 1.38e7T^{2} \)
67 \( 1 + 2.45e3T + 2.01e7T^{2} \)
71 \( 1 - 3.15e3T + 2.54e7T^{2} \)
73 \( 1 + 235. iT - 2.83e7T^{2} \)
79 \( 1 - 3.98e3T + 3.89e7T^{2} \)
83 \( 1 + 5.00e3iT - 4.74e7T^{2} \)
89 \( 1 - 7.60e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.25e4iT - 8.85e7T^{2} \)
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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

−12.93997637496691517006044544293, −12.28505991177933616882799617872, −11.06236510817886006001609349216, −9.857087987980905322922197116658, −9.009027270915914457101486548318, −8.032488596217999650160949835652, −6.20193349317751239318678132210, −4.98442354356402267929762590112, −3.90129006039695072896676767604, −1.79377021411836531981096570147, 0.71911200776598591929907872419, 2.56239798627585230310104891607, 4.22011040149898411990694176674, 6.03067691849715852510849060060, 7.33741622320023770826533690824, 7.67209360208109298291859642127, 9.598859300611313376729009545953, 10.58574090065378163520661760306, 11.53673190360222920459203613827, 12.80542608475762512480276720991

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