Properties

Degree $2$
Conductor $1512$
Sign $0.921 + 0.388i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.26 − 2.19i)5-s + (2.64 − 0.160i)7-s + (1.19 + 2.06i)11-s − 0.461·13-s + (3.64 + 6.30i)17-s + (4.32 − 7.48i)19-s + (−1.49 + 2.58i)23-s + (−0.721 − 1.24i)25-s − 3.55·29-s + (4.15 + 7.18i)31-s + (2.99 − 6.00i)35-s + (1.10 − 1.90i)37-s − 6.81·41-s + 2.38·43-s + (4.21 − 7.30i)47-s + ⋯
L(s)  = 1  + (0.567 − 0.983i)5-s + (0.998 − 0.0607i)7-s + (0.358 + 0.621i)11-s − 0.128·13-s + (0.883 + 1.52i)17-s + (0.991 − 1.71i)19-s + (−0.310 + 0.538i)23-s + (−0.144 − 0.249i)25-s − 0.660·29-s + (0.745 + 1.29i)31-s + (0.506 − 1.01i)35-s + (0.181 − 0.313i)37-s − 1.06·41-s + 0.363·43-s + (0.615 − 1.06i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.921 + 0.388i$
Motivic weight: \(1\)
Character: $\chi_{1512} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.921 + 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.264147774\)
\(L(\frac12)\) \(\approx\) \(2.264147774\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.160i)T \)
good5 \( 1 + (-1.26 + 2.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.19 - 2.06i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.461T + 13T^{2} \)
17 \( 1 + (-3.64 - 6.30i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.32 + 7.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.49 - 2.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.55T + 29T^{2} \)
31 \( 1 + (-4.15 - 7.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.10 + 1.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.81T + 41T^{2} \)
43 \( 1 - 2.38T + 43T^{2} \)
47 \( 1 + (-4.21 + 7.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.181 - 0.313i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.73 + 6.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.08 - 5.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.87 + 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.72T + 71T^{2} \)
73 \( 1 + (4.67 + 8.10i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.97 + 8.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.20T + 83T^{2} \)
89 \( 1 + (3.29 - 5.70i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−9.246336874604424887151233983322, −8.777499960193607706694349142390, −7.85604512919962892688222819780, −7.14287600117997241855574439718, −5.98342791165406855587889206392, −5.12273883320387160833441648150, −4.65350690897521298776286836587, −3.45005141234438452511195762171, −1.92774144951291400983682395395, −1.17925045362973926136832937671, 1.22228501766281975872403110307, 2.47507232106565847224502883662, 3.34697309509325148818584609938, 4.50621734304248538862176138009, 5.62804309503546859199231984345, 6.07002394715772798637859778248, 7.29157686146219055529956203785, 7.75702409751209202439241167682, 8.696650965632960218695774913859, 9.773505419541202903378659874836

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