Properties

Degree $2$
Conductor $1512$
Sign $0.984 + 0.174i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 0.633i)5-s + (2.12 + 1.57i)7-s + (1.92 − 3.33i)11-s − 4.87·13-s + (3.09 − 5.36i)17-s + (−1.39 − 2.41i)19-s + (3.69 + 6.40i)23-s + (2.23 − 3.86i)25-s + 9.78·29-s + (3.15 − 5.46i)31-s + (−0.216 + 1.92i)35-s + (2.82 + 4.88i)37-s + 1.50·41-s − 12.1·43-s + (−2.95 − 5.12i)47-s + ⋯
L(s)  = 1  + (0.163 + 0.283i)5-s + (0.804 + 0.593i)7-s + (0.580 − 1.00i)11-s − 1.35·13-s + (0.751 − 1.30i)17-s + (−0.320 − 0.555i)19-s + (0.771 + 1.33i)23-s + (0.446 − 0.773i)25-s + 1.81·29-s + (0.566 − 0.982i)31-s + (−0.0365 + 0.325i)35-s + (0.463 + 0.803i)37-s + 0.234·41-s − 1.85·43-s + (−0.431 − 0.747i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.984 + 0.174i$
Motivic weight: \(1\)
Character: $\chi_{1512} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.984 + 0.174i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.948348686\)
\(L(\frac12)\) \(\approx\) \(1.948348686\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 - 1.57i)T \)
good5 \( 1 + (-0.366 - 0.633i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.92 + 3.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.87T + 13T^{2} \)
17 \( 1 + (-3.09 + 5.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.39 + 2.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.69 - 6.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.78T + 29T^{2} \)
31 \( 1 + (-3.15 + 5.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.82 - 4.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.50T + 41T^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 + (2.95 + 5.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.43 - 11.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.19 + 5.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.29 - 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.834 - 1.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + (-0.720 + 1.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.55 + 6.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.98T + 83T^{2} \)
89 \( 1 + (-3.62 - 6.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−9.491732516731452842128066814704, −8.626110032368984417710659721359, −7.916518771780469003163511195091, −7.02739101585339592590459832995, −6.21800798363117205732611554704, −5.15270182521971114134182608190, −4.68583477894418457035811748085, −3.14872905933703799236455747580, −2.47839516799113917776833262370, −0.955432690791957019430356584792, 1.17362554084108333378873687218, 2.18490873170643708916146804000, 3.57111728208576798152175456248, 4.76900943358229985652075211389, 4.93027151709576715814182736068, 6.44670126482714588458723349131, 7.01427269828487735905521232420, 8.027491558147959941356724245165, 8.513893656570943940425545638726, 9.677682719711021045479136694054

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