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Properties

Label 2-1512-63.58-c1-0-6
Degree $2$
Conductor $1512$
Sign $-0.342 - 0.939i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.38 + 2.40i)5-s + (1.74 + 1.99i)7-s + (−1.71 + 2.97i)11-s + (−0.429 + 0.743i)13-s + (0.405 + 0.701i)17-s + (0.750 − 1.29i)19-s + (3.82 + 6.62i)23-s + (−1.34 + 2.32i)25-s + (−3.99 − 6.92i)29-s − 7.21·31-s + (−2.37 + 6.93i)35-s + (0.458 − 0.793i)37-s + (−1.67 + 2.90i)41-s + (1.20 + 2.08i)43-s + 0.615·47-s + ⋯
L(s)  = 1  + (0.619 + 1.07i)5-s + (0.657 + 0.753i)7-s + (−0.518 + 0.898i)11-s + (−0.119 + 0.206i)13-s + (0.0982 + 0.170i)17-s + (0.172 − 0.298i)19-s + (0.797 + 1.38i)23-s + (−0.268 + 0.464i)25-s + (−0.742 − 1.28i)29-s − 1.29·31-s + (−0.400 + 1.17i)35-s + (0.0753 − 0.130i)37-s + (−0.261 + 0.453i)41-s + (0.183 + 0.318i)43-s + 0.0897·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.777228329\)
\(L(\frac12)\) \(\approx\) \(1.777228329\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.74 - 1.99i)T \)
good5 \( 1 + (-1.38 - 2.40i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.71 - 2.97i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.429 - 0.743i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.405 - 0.701i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.750 + 1.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.82 - 6.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.99 + 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.21T + 31T^{2} \)
37 \( 1 + (-0.458 + 0.793i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.67 - 2.90i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.20 - 2.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.615T + 47T^{2} \)
53 \( 1 + (6.31 + 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.46T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 - 16.2T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (4.16 + 7.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 2.75T + 79T^{2} \)
83 \( 1 + (-5.75 - 9.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.11 + 8.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.82 - 6.63i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.672747482733727383320851191111, −9.140062958439838695050468094660, −7.936762751213177748231023756690, −7.34985481209011653434205401351, −6.48858709094897741717595115074, −5.56844783381921829717259112012, −4.92886077351963951964658372341, −3.60990276883599190352294569186, −2.49318030365010865437381055610, −1.82776566430708496835712152719, 0.69574623429451039035212415800, 1.74687120786957155140275687086, 3.13481254598460723384379065187, 4.28928639613196808961783554020, 5.19723286836683955771912153495, 5.63258362856342488049350458651, 6.89307752503672539919949459734, 7.69845244915836425149609224158, 8.609166419281366385368265202430, 9.006270891453828637856831302265

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