Properties

Label 2-112-1.1-c9-0-20
Degree $2$
Conductor $112$
Sign $-1$
Analytic cond. $57.6840$
Root an. cond. $7.59499$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 560·5-s + 2.40e3·7-s − 1.96e4·9-s + 5.41e4·11-s − 1.13e5·13-s + 3.36e3·15-s + 6.26e3·17-s − 2.57e5·19-s + 1.44e4·21-s + 2.66e5·23-s − 1.63e6·25-s − 2.35e5·27-s + 1.57e6·29-s + 4.63e6·31-s + 3.24e5·33-s + 1.34e6·35-s − 1.19e7·37-s − 6.79e5·39-s + 2.19e7·41-s − 2.75e7·43-s − 1.10e7·45-s − 5.29e7·47-s + 5.76e6·49-s + 3.75e4·51-s + 1.62e7·53-s + 3.03e7·55-s + ⋯
L(s)  = 1  + 0.0427·3-s + 0.400·5-s + 0.377·7-s − 0.998·9-s + 1.11·11-s − 1.09·13-s + 0.0171·15-s + 0.0181·17-s − 0.452·19-s + 0.0161·21-s + 0.198·23-s − 0.839·25-s − 0.0854·27-s + 0.413·29-s + 0.901·31-s + 0.0476·33-s + 0.151·35-s − 1.04·37-s − 0.0470·39-s + 1.21·41-s − 1.22·43-s − 0.399·45-s − 1.58·47-s + 1/7·49-s + 0.000777·51-s + 0.282·53-s + 0.446·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-1$
Analytic conductor: \(57.6840\)
Root analytic conductor: \(7.59499\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: $\chi_{112} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 112,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - p^{4} T \)
good3 \( 1 - 2 p T + p^{9} T^{2} \)
5 \( 1 - 112 p T + p^{9} T^{2} \)
11 \( 1 - 54152 T + p^{9} T^{2} \)
13 \( 1 + 113172 T + p^{9} T^{2} \)
17 \( 1 - 6262 T + p^{9} T^{2} \)
19 \( 1 + 257078 T + p^{9} T^{2} \)
23 \( 1 - 266000 T + p^{9} T^{2} \)
29 \( 1 - 1574714 T + p^{9} T^{2} \)
31 \( 1 - 4637484 T + p^{9} T^{2} \)
37 \( 1 + 11946238 T + p^{9} T^{2} \)
41 \( 1 - 21909126 T + p^{9} T^{2} \)
43 \( 1 + 27520592 T + p^{9} T^{2} \)
47 \( 1 + 52927836 T + p^{9} T^{2} \)
53 \( 1 - 16221222 T + p^{9} T^{2} \)
59 \( 1 - 140509618 T + p^{9} T^{2} \)
61 \( 1 + 202963560 T + p^{9} T^{2} \)
67 \( 1 + 153734572 T + p^{9} T^{2} \)
71 \( 1 + 3938816 p T + p^{9} T^{2} \)
73 \( 1 + 404022830 T + p^{9} T^{2} \)
79 \( 1 - 130689816 T + p^{9} T^{2} \)
83 \( 1 + 420134014 T + p^{9} T^{2} \)
89 \( 1 + 469542390 T + p^{9} T^{2} \)
97 \( 1 + 872501690 T + p^{9} T^{2} \)
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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

−11.49586291978623287930837181573, −10.19080233502646792323244937467, −9.166476156622793922479191888335, −8.168444791565768090803692462068, −6.80493805394212719255604861014, −5.68709168885975125727836716464, −4.44170219983506974825667342601, −2.89568038168686286300271099456, −1.61723214045253413550834420089, 0, 1.61723214045253413550834420089, 2.89568038168686286300271099456, 4.44170219983506974825667342601, 5.68709168885975125727836716464, 6.80493805394212719255604861014, 8.168444791565768090803692462068, 9.166476156622793922479191888335, 10.19080233502646792323244937467, 11.49586291978623287930837181573

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