Properties

Label 2-112-1.1-c9-0-0
Degree $2$
Conductor $112$
Sign $1$
Analytic cond. $57.6840$
Root an. cond. $7.59499$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 233.·3-s − 356.·5-s − 2.40e3·7-s + 3.46e4·9-s − 7.28e4·11-s + 3.93e4·13-s + 8.31e4·15-s − 5.11e5·17-s − 9.00e5·19-s + 5.59e5·21-s − 3.82e5·23-s − 1.82e6·25-s − 3.48e6·27-s − 4.73e6·29-s + 7.93e5·31-s + 1.69e7·33-s + 8.56e5·35-s − 1.72e7·37-s − 9.16e6·39-s − 1.53e7·41-s − 1.87e7·43-s − 1.23e7·45-s + 4.68e7·47-s + 5.76e6·49-s + 1.19e8·51-s + 2.06e7·53-s + 2.60e7·55-s + ⋯
L(s)  = 1  − 1.66·3-s − 0.255·5-s − 0.377·7-s + 1.75·9-s − 1.50·11-s + 0.382·13-s + 0.424·15-s − 1.48·17-s − 1.58·19-s + 0.627·21-s − 0.285·23-s − 0.934·25-s − 1.26·27-s − 1.24·29-s + 0.154·31-s + 2.49·33-s + 0.0964·35-s − 1.51·37-s − 0.634·39-s − 0.848·41-s − 0.834·43-s − 0.449·45-s + 1.40·47-s + 0.142·49-s + 2.46·51-s + 0.360·53-s + 0.383·55-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: no
Arithmetic: yes
Character: $\chi_{112} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.04850010179\)
\(L(\frac12)\) \(\approx\) \(0.04850010179\)
\(L(\frac{11}{2})\) 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 \( 1 + 2.40e3T \)
good3 \( 1 + 233.T + 1.96e4T^{2} \)
5 \( 1 + 356.T + 1.95e6T^{2} \)
11 \( 1 + 7.28e4T + 2.35e9T^{2} \)
13 \( 1 - 3.93e4T + 1.06e10T^{2} \)
17 \( 1 + 5.11e5T + 1.18e11T^{2} \)
19 \( 1 + 9.00e5T + 3.22e11T^{2} \)
23 \( 1 + 3.82e5T + 1.80e12T^{2} \)
29 \( 1 + 4.73e6T + 1.45e13T^{2} \)
31 \( 1 - 7.93e5T + 2.64e13T^{2} \)
37 \( 1 + 1.72e7T + 1.29e14T^{2} \)
41 \( 1 + 1.53e7T + 3.27e14T^{2} \)
43 \( 1 + 1.87e7T + 5.02e14T^{2} \)
47 \( 1 - 4.68e7T + 1.11e15T^{2} \)
53 \( 1 - 2.06e7T + 3.29e15T^{2} \)
59 \( 1 - 1.30e7T + 8.66e15T^{2} \)
61 \( 1 - 1.55e8T + 1.16e16T^{2} \)
67 \( 1 + 2.45e8T + 2.72e16T^{2} \)
71 \( 1 - 3.87e8T + 4.58e16T^{2} \)
73 \( 1 + 3.12e8T + 5.88e16T^{2} \)
79 \( 1 + 2.41e8T + 1.19e17T^{2} \)
83 \( 1 + 2.00e8T + 1.86e17T^{2} \)
89 \( 1 - 7.00e7T + 3.50e17T^{2} \)
97 \( 1 - 6.69e8T + 7.60e17T^{2} \)
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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

−11.70821549297374382700664845082, −10.81869814273198786265970996829, −10.23143695870121057083905311385, −8.578989212147553950771210624452, −7.14594433182344877090832915868, −6.16318604668279791165474188114, −5.20699523842490408126911721651, −4.05724606289318206373335108054, −2.06474294266778208537135673935, −0.12250774552239681175279965197, 0.12250774552239681175279965197, 2.06474294266778208537135673935, 4.05724606289318206373335108054, 5.20699523842490408126911721651, 6.16318604668279791165474188114, 7.14594433182344877090832915868, 8.578989212147553950771210624452, 10.23143695870121057083905311385, 10.81869814273198786265970996829, 11.70821549297374382700664845082

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