Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.620 + 0.784i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + (1.93 + 1.80i)7-s − 8-s + 3.87i·11-s + 1.60·13-s + (−1.93 − 1.80i)14-s + 16-s − 8.11i·17-s − 2.63i·19-s − 3.87i·22-s − 5.47·23-s − 1.60·26-s + (1.93 + 1.80i)28-s − 5.47i·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.732 + 0.681i)7-s − 0.353·8-s + 1.16i·11-s + 0.445·13-s + (−0.517 − 0.481i)14-s + 0.250·16-s − 1.96i·17-s − 0.605i·19-s − 0.825i·22-s − 1.14·23-s − 0.314·26-s + (0.366 + 0.340i)28-s − 1.01i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.620 + 0.784i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (3149, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.620 + 0.784i)$
$L(1)$  $\approx$  $1.223682531$
$L(\frac12)$  $\approx$  $1.223682531$
$L(\frac{3}{2})$   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 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-1.93 - 1.80i)T \)
good11 \( 1 - 3.87iT - 11T^{2} \)
13 \( 1 - 1.60T + 13T^{2} \)
17 \( 1 + 8.11iT - 17T^{2} \)
19 \( 1 + 2.63iT - 19T^{2} \)
23 \( 1 + 5.47T + 23T^{2} \)
29 \( 1 + 5.47iT - 29T^{2} \)
31 \( 1 + 3.73iT - 31T^{2} \)
37 \( 1 + 4.51iT - 37T^{2} \)
41 \( 1 + 1.60T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 + 11.1iT - 47T^{2} \)
53 \( 1 - 2.26T + 53T^{2} \)
59 \( 1 + 4.61T + 59T^{2} \)
61 \( 1 - 11.8iT - 61T^{2} \)
67 \( 1 + 6.90iT - 67T^{2} \)
71 \( 1 - 2.63iT - 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 - 8.01T + 79T^{2} \)
83 \( 1 - 3.20iT - 83T^{2} \)
89 \( 1 - 17.8T + 89T^{2} \)
97 \( 1 - 8.68T + 97T^{2} \)
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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

−8.683930898360714435751768626879, −7.76885446252274662300643666804, −7.32365548601391587273415903066, −6.47509475382714845995061365944, −5.49297620339672565989241376035, −4.84512513366339356179159237586, −3.85565412775405198212443918295, −2.45258268324297001481658316822, −2.03798989673770666836367162478, −0.53098921986875441939843042025, 1.11863467547930060166929756175, 1.81034895927463876159996144861, 3.28935920037945311442685445609, 3.90440265194172653559021790835, 4.98294481008181118096478608761, 6.13011554513155647229932354722, 6.35682395818301436912241857373, 7.61795014016908908532407959328, 8.216002848958300643993939539702, 8.475559110350237719847998299363

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