Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.911 + 0.411i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.849 + 1.47i)5-s + (−2.64 + 0.0963i)7-s + (−2.14 − 3.71i)11-s + 2.69·13-s + (1.64 + 2.84i)17-s + (1.79 − 3.10i)19-s + (1.29 − 2.24i)23-s + (1.05 + 1.82i)25-s − 7.68·29-s + (0.461 + 0.798i)31-s + (2.10 − 3.97i)35-s + (5.23 − 9.06i)37-s + 5.76·41-s + 11.0·43-s + (2.66 − 4.60i)47-s + ⋯
L(s)  = 1  + (−0.380 + 0.658i)5-s + (−0.999 + 0.0364i)7-s + (−0.646 − 1.11i)11-s + 0.748·13-s + (0.398 + 0.690i)17-s + (0.411 − 0.712i)19-s + (0.269 − 0.467i)23-s + (0.211 + 0.365i)25-s − 1.42·29-s + (0.0828 + 0.143i)31-s + (0.355 − 0.671i)35-s + (0.860 − 1.48i)37-s + 0.900·41-s + 1.69·43-s + (0.388 − 0.672i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.911 + 0.411i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (1297, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ 0.911 + 0.411i)\)
\(L(1)\)  \(\approx\)  \(1.251198732\)
\(L(\frac12)\)  \(\approx\)  \(1.251198732\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (2.64 - 0.0963i)T \)
good5 \( 1 + (0.849 - 1.47i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.14 + 3.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.69T + 13T^{2} \)
17 \( 1 + (-1.64 - 2.84i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.79 + 3.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.29 + 2.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.68T + 29T^{2} \)
31 \( 1 + (-0.461 - 0.798i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.23 + 9.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.76T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + (-2.66 + 4.60i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.23 - 3.87i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.60 - 7.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.08 + 7.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.81 - 3.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + (0.333 + 0.576i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.88 + 8.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.25T + 83T^{2} \)
89 \( 1 + (0.800 - 1.38i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.823T + 97T^{2} \)
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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

−9.276669485182934587391494643474, −8.760808372838730569269788356291, −7.66617191249431493443502782824, −7.10351660236472333474662362933, −5.99382622649620038619253364111, −5.64778626781019685013820108424, −4.05737355081620069039807099690, −3.35216423536734408374929232604, −2.55526998090462977539079991872, −0.65249653878592473157851141255, 0.961122134908316418280108868256, 2.48549531242273983748690933330, 3.57864529782449982388119905103, 4.44024066074217019992901117038, 5.41098488746582213165435274190, 6.20563846561401234576997781679, 7.30980877092455024764784349021, 7.78559196469087831814778049805, 8.826782777138938712758455165949, 9.592408739656214487476360432272

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