Properties

Label 2-112-1.1-c5-0-13
Degree $2$
Conductor $112$
Sign $-1$
Analytic cond. $17.9629$
Root an. cond. $4.23827$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·3-s − 56·5-s + 49·7-s − 47·9-s − 232·11-s − 140·13-s − 784·15-s − 1.72e3·17-s + 98·19-s + 686·21-s − 1.82e3·23-s + 11·25-s − 4.06e3·27-s + 3.41e3·29-s + 7.64e3·31-s − 3.24e3·33-s − 2.74e3·35-s − 1.03e4·37-s − 1.96e3·39-s − 1.79e4·41-s − 1.08e4·43-s + 2.63e3·45-s − 9.32e3·47-s + 2.40e3·49-s − 2.41e4·51-s + 2.26e3·53-s + 1.29e4·55-s + ⋯
L(s)  = 1  + 0.898·3-s − 1.00·5-s + 0.377·7-s − 0.193·9-s − 0.578·11-s − 0.229·13-s − 0.899·15-s − 1.44·17-s + 0.0622·19-s + 0.339·21-s − 0.718·23-s + 0.00351·25-s − 1.07·27-s + 0.754·29-s + 1.42·31-s − 0.519·33-s − 0.378·35-s − 1.24·37-s − 0.206·39-s − 1.66·41-s − 0.897·43-s + 0.193·45-s − 0.615·47-s + 1/7·49-s − 1.29·51-s + 0.110·53-s + 0.579·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-1$
Analytic conductor: \(17.9629\)
Root analytic conductor: \(4.23827\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 112,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - p^{2} T \)
good3 \( 1 - 14 T + p^{5} T^{2} \)
5 \( 1 + 56 T + p^{5} T^{2} \)
11 \( 1 + 232 T + p^{5} T^{2} \)
13 \( 1 + 140 T + p^{5} T^{2} \)
17 \( 1 + 1722 T + p^{5} T^{2} \)
19 \( 1 - 98 T + p^{5} T^{2} \)
23 \( 1 + 1824 T + p^{5} T^{2} \)
29 \( 1 - 3418 T + p^{5} T^{2} \)
31 \( 1 - 7644 T + p^{5} T^{2} \)
37 \( 1 + 10398 T + p^{5} T^{2} \)
41 \( 1 + 17962 T + p^{5} T^{2} \)
43 \( 1 + 10880 T + p^{5} T^{2} \)
47 \( 1 + 9324 T + p^{5} T^{2} \)
53 \( 1 - 2262 T + p^{5} T^{2} \)
59 \( 1 - 2730 T + p^{5} T^{2} \)
61 \( 1 - 25648 T + p^{5} T^{2} \)
67 \( 1 - 48404 T + p^{5} T^{2} \)
71 \( 1 - 58560 T + p^{5} T^{2} \)
73 \( 1 - 68082 T + p^{5} T^{2} \)
79 \( 1 + 31784 T + p^{5} T^{2} \)
83 \( 1 - 20538 T + p^{5} T^{2} \)
89 \( 1 + 50582 T + p^{5} T^{2} \)
97 \( 1 + 58506 T + p^{5} T^{2} \)
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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

−12.06445015013949851453999474638, −11.24817683734094753711936799202, −9.966773655781626956546010508595, −8.483822123022907380484465042216, −8.131547140187430213274395118746, −6.77409075466514291700472522927, −4.91537525656401637745330793469, −3.62299926268921600296018605877, −2.27647681398025495199245603775, 0, 2.27647681398025495199245603775, 3.62299926268921600296018605877, 4.91537525656401637745330793469, 6.77409075466514291700472522927, 8.131547140187430213274395118746, 8.483822123022907380484465042216, 9.966773655781626956546010508595, 11.24817683734094753711936799202, 12.06445015013949851453999474638

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