Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $-0.100 + 0.994i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 2.36i)5-s + (−0.931 − 2.47i)7-s + (−1.17 − 2.04i)11-s + 6.68·13-s + (2.09 + 3.63i)17-s + (−1.80 + 3.11i)19-s + (3.40 − 5.90i)23-s + (−1.23 − 2.13i)25-s − 1.00·29-s + (−1.05 − 1.82i)31-s + (−7.13 − 1.17i)35-s + (2.47 − 4.29i)37-s − 8.91·41-s − 2.25·43-s + (5.85 − 10.1i)47-s + ⋯
L(s)  = 1  + (0.610 − 1.05i)5-s + (−0.352 − 0.935i)7-s + (−0.355 − 0.615i)11-s + 1.85·13-s + (0.508 + 0.881i)17-s + (−0.413 + 0.715i)19-s + (0.710 − 1.23i)23-s + (−0.246 − 0.426i)25-s − 0.186·29-s + (−0.189 − 0.327i)31-s + (−1.20 − 0.199i)35-s + (0.407 − 0.706i)37-s − 1.39·41-s − 0.343·43-s + (0.853 − 1.47i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.100 + 0.994i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (1297, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ -0.100 + 0.994i)\)
\(L(1)\)  \(\approx\)  \(1.823634822\)
\(L(\frac12)\)  \(\approx\)  \(1.823634822\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.931 + 2.47i)T \)
good5 \( 1 + (-1.36 + 2.36i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.17 + 2.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.68T + 13T^{2} \)
17 \( 1 + (-2.09 - 3.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.80 - 3.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.40 + 5.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.00T + 29T^{2} \)
31 \( 1 + (1.05 + 1.82i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.47 + 4.29i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.91T + 41T^{2} \)
43 \( 1 + 2.25T + 43T^{2} \)
47 \( 1 + (-5.85 + 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.13 + 7.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.91 + 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.00 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.03 - 5.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
73 \( 1 + (-2.28 - 3.96i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.252 + 0.437i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + (-4.22 + 7.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.25T + 97T^{2} \)
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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

−9.069116934390451538202198439308, −8.546742573271565045332257533143, −7.900958357438526743789831775284, −6.61022608723782466511133329170, −6.00852873620388987031070609012, −5.19203988481934877810782024758, −4.05865006999489004015271121179, −3.40668487466838139521213483374, −1.71928211349230438129959879187, −0.76507829935207486510080387595, 1.58385892550694854229078788275, 2.80316610804178853322130161265, 3.34197672292601282561534931894, 4.81615644846184978034655805030, 5.77213936695705964563659030626, 6.37626717622924025223013529236, 7.12268924800866344610582736390, 8.092732873931718515467036708237, 9.132260860494195641033296896888, 9.520195882162496892058663335253

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