Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.654 - 0.755i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·2-s − 1.73i·3-s − 0.999·4-s + 5-s + 2.99·6-s + (2 + 1.73i)7-s + 1.73i·8-s − 2.99·9-s + 1.73i·10-s − 3.46i·11-s + 1.73i·12-s + (−2.99 + 3.46i)14-s − 1.73i·15-s − 5·16-s − 6·17-s − 5.19i·18-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.999i·3-s − 0.499·4-s + 0.447·5-s + 1.22·6-s + (0.755 + 0.654i)7-s + 0.612i·8-s − 0.999·9-s + 0.547i·10-s − 1.04i·11-s + 0.499i·12-s + (−0.801 + 0.925i)14-s − 0.447i·15-s − 1.25·16-s − 1.45·17-s − 1.22i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.654 - 0.755i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (41, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.654 - 0.755i)$
$L(1)$  $\approx$  $1.02762 + 0.469472i$
$L(\frac12)$  $\approx$  $1.02762 + 0.469472i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 - T \)
7 \( 1 + (-2 - 1.73i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−13.85073546011936671499247282926, −13.39455584922365110905441252970, −11.77580529432819705656530642586, −11.09076564889229001503688653183, −8.888675227619284864770247112347, −8.331081264168999936219915918912, −7.08173325171240219905876961637, −6.12624780347219580393116659172, −5.19569873787941261107632649192, −2.30989461230611783328334420826, 2.12350520376179601463862809320, 3.89188085758447586938992309253, 4.90641769224314713543468516814, 6.84319041448808681405548349722, 8.635844725493169134753519723351, 9.811044810689386859596554820332, 10.51196380823766213115919406936, 11.24803574257886029889692469881, 12.33787647325264601195125203552, 13.51046139945140927605779713303

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