Properties

Label 2-1512-63.16-c1-0-16
Degree $2$
Conductor $1512$
Sign $0.996 - 0.0815i$
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  + 3.52·5-s + (1.16 + 2.37i)7-s + 2.32·11-s + (−2.35 − 4.08i)13-s + (0.636 + 1.10i)17-s + (2.78 − 4.82i)19-s + 3.29·23-s + 7.45·25-s + (4.32 − 7.48i)29-s + (−4.25 + 7.37i)31-s + (4.11 + 8.38i)35-s + (−2.84 + 4.91i)37-s + (−1.66 − 2.88i)41-s + (0.0444 − 0.0769i)43-s + (3.52 + 6.10i)47-s + ⋯
L(s)  = 1  + 1.57·5-s + (0.441 + 0.897i)7-s + 0.699·11-s + (−0.654 − 1.13i)13-s + (0.154 + 0.267i)17-s + (0.638 − 1.10i)19-s + 0.687·23-s + 1.49·25-s + (0.802 − 1.38i)29-s + (−0.764 + 1.32i)31-s + (0.696 + 1.41i)35-s + (−0.466 + 0.808i)37-s + (−0.260 − 0.450i)41-s + (0.00677 − 0.0117i)43-s + (0.514 + 0.890i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.996 - 0.0815i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.996 - 0.0815i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.505727635\)
\(L(\frac12)\) \(\approx\) \(2.505727635\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.16 - 2.37i)T \)
good5 \( 1 - 3.52T + 5T^{2} \)
11 \( 1 - 2.32T + 11T^{2} \)
13 \( 1 + (2.35 + 4.08i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.636 - 1.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.78 + 4.82i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.29T + 23T^{2} \)
29 \( 1 + (-4.32 + 7.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.25 - 7.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.84 - 4.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.66 + 2.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0444 + 0.0769i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.52 - 6.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.41 + 5.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.99 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.67 + 11.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.06 - 5.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.30T + 71T^{2} \)
73 \( 1 + (-6.64 - 11.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.01 - 8.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.90 + 10.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.561 - 0.972i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.50 - 6.07i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.412079725345601429769341497578, −8.918292526582738234861690195482, −7.992641513574706866967293807802, −6.89533485016084737427332959033, −6.12524384154149408813131962110, −5.33041557444654715814454242822, −4.84110862332050664041532508020, −3.11457499565901127129046243009, −2.36290452416616982150906369192, −1.25236861432813743109547232837, 1.30669829559258387089132237429, 2.04559900240168328513458737872, 3.42718602808413399025714483896, 4.53809947467102750428862643912, 5.32794302894566850130363361990, 6.24755141163019951908843646275, 6.98114173680744758764847814299, 7.70296150280021275971311835143, 9.055091085116038684317106470515, 9.359879626900771730653034511495

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