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

\(L(1)\) \(\approx\) \(1.595243822\)
\(L(\frac12)\) \(\approx\) \(1.595243822\)
\(L(\frac{3}{2})\) 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

−9.258417815904829315638602300949, −8.675739765966401635850107051013, −7.56916305104901833324204133098, −7.21961639876473682881360663325, −6.33298385099311344319277814248, −5.00179160680759995384408334272, −4.59510871728247353361114770023, −3.08211060492443229169983418126, −2.60513839474922914792990708937, −0.75525348140570379484882994208, 1.16334916977505020075731593821, 2.18196983713344862654420760855, 3.81715834443573674442217148296, 4.46168586936407861509670770836, 5.26549507400089882584318309572, 6.12536980523087890390954710175, 7.37969644560365982563317397329, 8.187976674137681131331922819403, 8.354759148739930364926150081995, 9.522514508119975850109922157129

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