Properties

Degree 2
Conductor $ 5 \cdot 13 \cdot 31 $
Sign $-0.707 + 0.707i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·3-s − 4-s + i·5-s + 1.00·9-s + 1.41·12-s + (−0.707 + 0.707i)13-s − 1.41i·15-s + 16-s − 1.41·17-s + 2i·19-s i·20-s + 1.41·23-s − 25-s i·31-s − 1.00·36-s − 1.41i·37-s + ⋯
L(s)  = 1  − 1.41·3-s − 4-s + i·5-s + 1.00·9-s + 1.41·12-s + (−0.707 + 0.707i)13-s − 1.41i·15-s + 16-s − 1.41·17-s + 2i·19-s i·20-s + 1.41·23-s − 25-s i·31-s − 1.00·36-s − 1.41i·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2015\)    =    \(5 \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $-0.707 + 0.707i$
motivic weight  =  \(0\)
character  :  $\chi_{2015} (2014, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 2015,\ (\ :0),\ -0.707 + 0.707i)$
$L(\frac{1}{2})$  $\approx$  $0.03519285380$
$L(\frac12)$  $\approx$  $0.03519285380$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;13,\;31\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 - iT \)
13 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + iT \)
good2 \( 1 + T^{2} \)
3 \( 1 + 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 - 2iT - T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.41T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.839950893161899092385783196996, −9.312501696837907515997710166324, −8.204446814130813598681554664878, −7.32111852431977581249116548381, −6.49860957369408700507686464463, −5.93003974824987697273674769902, −5.01393982702516966652878955767, −4.34156489390107871611728109321, −3.36956349871373830768570155578, −1.85630368309975487466113369917, 0.03629652390274289110268692307, 1.14004336220876583025255489716, 2.97165447919623094562908197246, 4.48652628543504515649648409540, 4.98335495843254955774097140474, 5.18673344248147815268195993086, 6.42485092721915000073327158869, 7.08366216075101416046585234512, 8.321227114128265199325494107183, 8.871502481159826766475230748756

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