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  =  $(2,\ 2015,\ (\ :0),\ 0.707 - 0.707i)$
$L(\frac{1}{2})$  $\approx$  $1.529637734$
$L(\frac12)$  $\approx$  $1.529637734$
$L(1)$   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

−9.526048227715514060879179391755, −8.454569638432037948416064631025, −7.905929419800200784613826461383, −7.67248619175886183864425318000, −6.10943406679652768348230402057, −5.62668873291195176499693562043, −4.04180061169350097558750431173, −3.62453946667223181540704568407, −2.89908403831764470521292838867, −1.61494921427731080941020762608, 1.11190893893154543913427395402, 2.38982735964998717767716201328, 3.61139618957688211556248202949, 4.12786225794346189229918619943, 5.04812650819212702495375033823, 5.88231500710522946659600845852, 7.25004716080695604846422553132, 8.021348505173509776035545632566, 8.587365153684021298219123018196, 9.175789832161759726553532010418

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