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Properties

Label 2-1512-63.4-c1-0-9
Degree $2$
Conductor $1512$
Sign $0.281 - 0.959i$
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  + 2.52·5-s + (−1.07 + 2.41i)7-s + 5.71·11-s + (−2.45 + 4.24i)13-s + (−2.49 + 4.32i)17-s + (−0.00383 − 0.00664i)19-s − 0.667·23-s + 1.36·25-s + (−3.85 − 6.66i)29-s + (3.88 + 6.72i)31-s + (−2.71 + 6.09i)35-s + (−3.19 − 5.53i)37-s + (−5.21 + 9.02i)41-s + (4.42 + 7.67i)43-s + (1.08 − 1.87i)47-s + ⋯
L(s)  = 1  + 1.12·5-s + (−0.407 + 0.913i)7-s + 1.72·11-s + (−0.680 + 1.17i)13-s + (−0.605 + 1.04i)17-s + (−0.000880 − 0.00152i)19-s − 0.139·23-s + 0.273·25-s + (−0.715 − 1.23i)29-s + (0.697 + 1.20i)31-s + (−0.459 + 1.03i)35-s + (−0.525 − 0.909i)37-s + (−0.813 + 1.40i)41-s + (0.675 + 1.16i)43-s + (0.157 − 0.272i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.281 - 0.959i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.281 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.931016961\)
\(L(\frac12)\) \(\approx\) \(1.931016961\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.07 - 2.41i)T \)
good5 \( 1 - 2.52T + 5T^{2} \)
11 \( 1 - 5.71T + 11T^{2} \)
13 \( 1 + (2.45 - 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.49 - 4.32i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.00383 + 0.00664i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.667T + 23T^{2} \)
29 \( 1 + (3.85 + 6.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.88 - 6.72i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.19 + 5.53i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.21 - 9.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.42 - 7.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.08 + 1.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.69 + 6.40i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.261 + 0.453i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.49 + 7.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.54 - 4.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.68T + 71T^{2} \)
73 \( 1 + (1.52 - 2.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.08 - 5.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.258 - 0.448i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.19 + 2.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.32 - 7.49i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.529484750554784957477975118716, −9.066260621424866543537809318998, −8.295604435802723212488233263209, −6.81341073932875931145498863820, −6.42783432998205898408792259264, −5.74183158100597725947147929497, −4.61805778106561553861448520765, −3.68252775511603353424566641322, −2.28903564631101446945464213653, −1.64834067812789938757420693143, 0.76476763806817293118257717398, 2.03892361748427900166445118515, 3.23164112852879286775022179269, 4.19690560218956964437235720725, 5.24633035651399927964726175516, 6.08780538262587814847665018127, 6.90949727497633981208016287989, 7.44775744348940233562241695584, 8.786927350274836826492136729892, 9.377337423724430126850833316423

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