Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 $
Sign $0.894 + 0.447i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + i·3-s − 4-s + (−2 − i)5-s + 6-s + 2i·7-s + i·8-s − 9-s + (−1 + 2i)10-s + 2·11-s i·12-s − 6i·13-s + 2·14-s + (1 − 2i)15-s + 16-s + 2i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 0.408·6-s + 0.755i·7-s + 0.353i·8-s − 0.333·9-s + (−0.316 + 0.632i)10-s + 0.603·11-s − 0.288i·12-s − 1.66i·13-s + 0.534·14-s + (0.258 − 0.516i)15-s + 0.250·16-s + 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30\)    =    \(2 \cdot 3 \cdot 5\)
\( \varepsilon \)  =  $0.894 + 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{30} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 30,\ (\ :1/2),\ 0.894 + 0.447i)$
$L(1)$  $\approx$  $0.612627 - 0.144621i$
$L(\frac12)$  $\approx$  $0.612627 - 0.144621i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + iT \)
3 \( 1 - iT \)
5 \( 1 + (2 + i)T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−17.01977398561147585569982166153, −15.63832276156840719790648214054, −14.82613947152933352288907316812, −12.96250512409104260660029756949, −11.95451807699588727491650213315, −10.79733812128479603885789976626, −9.291364285307452496699123458412, −8.085457671479813666280641047905, −5.35115232455633761080372105825, −3.56141683746843219707232773454, 4.16710137136039276355392879535, 6.63650189600152426497619539048, 7.48664995322902406921466101924, 9.062236764009944146012759381636, 11.03406984828253276966084497277, 12.26964569530337323975039188135, 13.88854108804486098115315613885, 14.60665231932445477324661417962, 16.14791334000609191876902356433, 16.96826441731702099101614986322

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