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Properties

Label 2-1512-21.2-c0-0-1
Degree $2$
Conductor $1512$
Sign $-0.997 + 0.0633i$
Analytic cond. $0.754586$
Root an. cond. $0.868669$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)5-s + (−0.5 − 0.866i)7-s + (−1.22 + 0.707i)17-s + (−0.5 + 0.866i)19-s + (−1.22 − 0.707i)23-s + (0.499 + 0.866i)25-s + 1.41i·29-s + (−0.5 − 0.866i)31-s + 1.41i·35-s − 1.41i·41-s − 43-s + (−1.22 − 0.707i)47-s + (−0.499 + 0.866i)49-s + (1.22 − 0.707i)53-s + (−0.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)5-s + (−0.5 − 0.866i)7-s + (−1.22 + 0.707i)17-s + (−0.5 + 0.866i)19-s + (−1.22 − 0.707i)23-s + (0.499 + 0.866i)25-s + 1.41i·29-s + (−0.5 − 0.866i)31-s + 1.41i·35-s − 1.41i·41-s − 43-s + (−1.22 − 0.707i)47-s + (−0.499 + 0.866i)49-s + (1.22 − 0.707i)53-s + (−0.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.997 + 0.0633i$
Analytic conductor: \(0.754586\)
Root analytic conductor: \(0.868669\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :0),\ -0.997 + 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2300220928\)
\(L(\frac12)\) \(\approx\) \(0.2300220928\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.074830753296028819792620396905, −8.434421188555495422250046711256, −7.77525509038380861539430932005, −6.92461119943155048315300655522, −6.13291220823556469423796364741, −4.87141283895998802550395313181, −4.02266228333766174654521631316, −3.61233510932103958253907169298, −1.89125043096980056946884646816, −0.17344556651139102933713112430, 2.27137207591839449645698756223, 3.13747056925689054751223392936, 4.07885696346475108087784249844, 4.98809290396899686207334730754, 6.23098899497012005065889719163, 6.77828824004220703879908374950, 7.69254309801200419555736199604, 8.398419827036235140721738447559, 9.237885691877970537680944870884, 9.983653567746084048049611642881

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