Properties

Degree 2
Conductor 113
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s − 4-s + 2·5-s − 2·6-s + 3·8-s + 9-s − 2·10-s − 2·12-s + 2·13-s + 4·15-s − 16-s − 6·17-s − 18-s + 6·19-s − 2·20-s − 6·23-s + 6·24-s − 25-s − 2·26-s − 4·27-s − 6·29-s − 4·30-s − 4·31-s − 5·32-s + 6·34-s − 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.894·5-s − 0.816·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.577·12-s + 0.554·13-s + 1.03·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.37·19-s − 0.447·20-s − 1.25·23-s + 1.22·24-s − 1/5·25-s − 0.392·26-s − 0.769·27-s − 1.11·29-s − 0.730·30-s − 0.718·31-s − 0.883·32-s + 1.02·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(113\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{113} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 113,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.009184946$
$L(\frac12)$  $\approx$  $1.009184946$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 113$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 113$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad113 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.31390902194407, −18.14374566247207, −17.86141654658866, −16.66208060410326, −15.56409213262675, −14.28464892230888, −13.70413845073004, −13.08030829164543, −11.26364751904647, −9.938667014422479, −9.263870633435132, −8.504919444417912, −7.416493906175945, −5.665854248740159, −3.904140667768108, −2.052179095726990, 2.052179095726990, 3.904140667768108, 5.665854248740159, 7.416493906175945, 8.504919444417912, 9.263870633435132, 9.938667014422479, 11.26364751904647, 13.08030829164543, 13.70413845073004, 14.28464892230888, 15.56409213262675, 16.66208060410326, 17.86141654658866, 18.14374566247207, 19.31390902194407

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