Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.447 - 0.894i$
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 − 4-s + i·7-s + i·8-s + 4·11-s + 6i·13-s + 14-s + 16-s + 4i·17-s − 6·19-s − 4i·22-s + 6·26-s i·28-s − 6·29-s − 4·31-s i·32-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.377i·7-s + 0.353i·8-s + 1.20·11-s + 1.66i·13-s + 0.267·14-s + 0.250·16-s + 0.970i·17-s − 1.37·19-s − 0.852i·22-s + 1.17·26-s − 0.188i·28-s − 1.11·29-s − 0.718·31-s − 0.176i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.447 - 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.447 - 0.894i)$
$L(1)$  $\approx$  $0.5982553931$
$L(\frac12)$  $\approx$  $0.5982553931$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - iT \)
good11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 10iT - 47T^{2} \)
53 \( 1 + 14iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.989944523851944745677400490040, −8.551688120957419184863446133736, −7.43204772394178331423441551924, −6.51695129391570008242047856145, −6.03471319516934027341951167081, −4.84336855542512028900222161397, −4.04741547347709638169950482970, −3.55794752597154190458651212811, −2.02369591564810194406885870016, −1.69788553547637432017326320021, 0.18014242094875388600744436718, 1.47430329652078000632173265111, 2.95592546566307436550590342399, 3.80749737426478249365233225326, 4.64343749736367863339587409196, 5.49338183265968951560534288483, 6.21935800601662952174837603363, 6.99427010476788893980956406078, 7.57417570802799638399813567576, 8.452638401522782125996941855119

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