Properties

Degree 2
Conductor $ 3 \cdot 13 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s + 9-s + 12-s − 13-s + 16-s − 25-s − 27-s − 36-s + 39-s + 2·43-s − 48-s + 49-s + 52-s − 2·61-s − 64-s + 75-s − 2·79-s + 81-s + 100-s + 2·103-s + 108-s − 117-s + ⋯
L(s)  = 1  − 3-s − 4-s + 9-s + 12-s − 13-s + 16-s − 25-s − 27-s − 36-s + 39-s + 2·43-s − 48-s + 49-s + 52-s − 2·61-s − 64-s + 75-s − 2·79-s + 81-s + 100-s + 2·103-s + 108-s − 117-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(39\)    =    \(3 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{39} (38, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 39,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.2913537228$
$L(\frac12)$  $\approx$  $0.2913537228$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;13\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )^{2} \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 + T )^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−16.89615994646558642421555820819, −15.51291484878802936574704349484, −14.14821592234820384952611063785, −12.89937159831719324582676854317, −11.93271353025467403728401266104, −10.42608525426608152464833032909, −9.334261916221688779645422148440, −7.54499009544086055036625670577, −5.72936210305730426417401616626, −4.39582175376675801531882680457, 4.39582175376675801531882680457, 5.72936210305730426417401616626, 7.54499009544086055036625670577, 9.334261916221688779645422148440, 10.42608525426608152464833032909, 11.93271353025467403728401266104, 12.89937159831719324582676854317, 14.14821592234820384952611063785, 15.51291484878802936574704349484, 16.89615994646558642421555820819

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