Properties

Degree $2$
Conductor $1512$
Sign $0.292 - 0.956i$
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 + (0.431 + 2.61i)7-s + (2.54 + 4.40i)11-s − 3.48·13-s + (2.09 + 3.63i)17-s + (−3.16 + 5.47i)19-s + (−3.04 + 5.26i)23-s + (−1.23 − 2.13i)25-s − 6.45·29-s + (−4.77 − 8.27i)31-s + (6.76 + 2.54i)35-s + (0.118 − 0.205i)37-s + 5.98·41-s + 12.6·43-s + (−4.31 + 7.48i)47-s + ⋯
L(s)  = 1  + (0.610 − 1.05i)5-s + (0.163 + 0.986i)7-s + (0.767 + 1.32i)11-s − 0.967·13-s + (0.508 + 0.881i)17-s + (−0.725 + 1.25i)19-s + (−0.634 + 1.09i)23-s + (−0.246 − 0.426i)25-s − 1.19·29-s + (−0.857 − 1.48i)31-s + (1.14 + 0.430i)35-s + (0.0195 − 0.0337i)37-s + 0.934·41-s + 1.92·43-s + (−0.629 + 1.09i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.292 - 0.956i$
Motivic weight: \(1\)
Character: $\chi_{1512} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.292 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.604235213\)
\(L(\frac12)\) \(\approx\) \(1.604235213\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.431 - 2.61i)T \)
good5 \( 1 + (-1.36 + 2.36i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.54 - 4.40i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.48T + 13T^{2} \)
17 \( 1 + (-2.09 - 3.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.16 - 5.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.04 - 5.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.45T + 29T^{2} \)
31 \( 1 + (4.77 + 8.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.118 + 0.205i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.98T + 41T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 + (4.31 - 7.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.30 - 4.00i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.18 + 3.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.443 + 0.768i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.39 - 7.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.55T + 71T^{2} \)
73 \( 1 + (6.52 + 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.61 + 4.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.73T + 83T^{2} \)
89 \( 1 + (-1.50 + 2.60i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.338427026359203353656721821277, −9.246097291539211244576489965261, −7.974224940243021292616791174144, −7.46080507001994099118213704270, −5.94603813061307649472675820999, −5.74116764658187656901783772588, −4.63322400834819955110702089601, −3.87150878006570065046378251233, −2.13176396265110734717360686173, −1.65227455117624153009864282352, 0.61579662651317965462772858530, 2.25238859790963106480688488252, 3.16064575176413229043287190986, 4.12642407853949344325560486228, 5.21240873686583979130617044318, 6.21726411477842568509616805127, 6.92438211856721495296582141849, 7.43995525782724173810358205125, 8.602837959485299291385136810508, 9.385633523099047230157066195502

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