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

\(L(1)\) \(\approx\) \(0.3819104545\)
\(L(\frac12)\) \(\approx\) \(0.3819104545\)
\(L(\frac{3}{2})\) 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

−9.109004387137296116219387963508, −8.378496104096835023514896753099, −7.69291399010031644853424145756, −6.82408887792575727700401582464, −5.97293005771194312395671668980, −5.18559990834965709586761057918, −4.04602859476724519563555549999, −2.86767121388427997037417498206, −2.43001701036444273426795257735, −0.15341856744821742687036161406, 1.30514332682071178552361819466, 2.62117604846453655452189502923, 4.08675172015537245675401755310, 4.66514392604151994707712716409, 5.21157142086493921837784220978, 6.79122873450382175999320980292, 7.31698104003392194328232335546, 8.060754498352380529305122331599, 8.889649881331656869419661815909, 9.652677123626499364204669397953

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