Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.539 - 0.841i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.63 + 0.189i)7-s − 0.999·8-s + (−2.55 + 1.47i)11-s − 3.93·13-s + (1.48 − 2.19i)14-s + (−0.5 + 0.866i)16-s + (−0.346 + 0.199i)17-s + (−0.0305 − 0.0176i)19-s + 2.94i·22-s + (−1.86 + 3.23i)23-s + (−1.96 + 3.40i)26-s + (−1.15 − 2.38i)28-s + 8.89i·29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.997 + 0.0716i)7-s − 0.353·8-s + (−0.769 + 0.444i)11-s − 1.09·13-s + (0.396 − 0.585i)14-s + (−0.125 + 0.216i)16-s + (−0.0839 + 0.0484i)17-s + (−0.00700 − 0.00404i)19-s + 0.628i·22-s + (−0.389 + 0.673i)23-s + (−0.385 + 0.667i)26-s + (−0.218 − 0.449i)28-s + 1.65i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.539 - 0.841i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.539 - 0.841i)$
$L(1)$  $\approx$  $1.363091286$
$L(\frac12)$  $\approx$  $1.363091286$
$L(\frac{3}{2})$   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 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.63 - 0.189i)T \)
good11 \( 1 + (2.55 - 1.47i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.93T + 13T^{2} \)
17 \( 1 + (0.346 - 0.199i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0305 + 0.0176i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.86 - 3.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.89iT - 29T^{2} \)
31 \( 1 + (-0.717 + 0.414i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.86 - 3.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.31T + 41T^{2} \)
43 \( 1 - 3.03iT - 43T^{2} \)
47 \( 1 + (-5.02 - 2.90i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.14 + 3.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.78 - 4.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.97 + 5.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.8 - 6.25i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.93iT - 71T^{2} \)
73 \( 1 + (-0.171 - 0.297i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.15 + 7.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (3.08 - 5.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−8.923326729918991889550454215072, −7.934857537383873684965645645840, −7.50196331469359380849001289312, −6.49456360533358781512079046271, −5.37352358448210351144066287948, −4.96350906333821255304220181228, −4.23273508911215594892795001242, −3.06986145795823201279516495328, −2.25692920006159304473315372718, −1.34004454611283292087291896969, 0.36280820123741468153197208753, 2.07269406179304581366334927281, 2.88630636271534328618609698983, 4.18689847081255272019214828809, 4.70664645692906874604597900317, 5.52286486806947220554163214014, 6.16903050667175581407931089613, 7.23344243377741927635123157495, 7.76428230074117342252835107568, 8.311674328166475589409190015847

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