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Properties

Label 2-1512-63.25-c1-0-12
Degree $2$
Conductor $1512$
Sign $0.574 + 0.818i$
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  + (−1.76 + 3.05i)5-s + (−2.63 + 0.176i)7-s + (−1.16 − 2.00i)11-s + (−2.35 − 4.08i)13-s + (0.636 − 1.10i)17-s + (2.78 + 4.82i)19-s + (−1.64 + 2.85i)23-s + (−3.72 − 6.45i)25-s + (4.32 − 7.48i)29-s + 8.51·31-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 − 7.05·47-s + ⋯
L(s)  = 1  + (−0.789 + 1.36i)5-s + (−0.997 + 0.0666i)7-s + (−0.349 − 0.605i)11-s + (−0.654 − 1.13i)13-s + (0.154 − 0.267i)17-s + (0.638 + 1.10i)19-s + (−0.343 + 0.595i)23-s + (−0.745 − 1.29i)25-s + (0.802 − 1.38i)29-s + 1.52·31-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 − 1.02·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7609395998\)
\(L(\frac12)\) \(\approx\) \(0.7609395998\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 0.176i)T \)
good5 \( 1 + (1.76 - 3.05i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.16 + 2.00i)T + (-5.5 + 9.52i)T^{2} \)
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 \( 1 + (1.64 - 2.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.32 + 7.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.51T + 31T^{2} \)
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 \( 1 + 7.05T + 47T^{2} \)
53 \( 1 + (3.41 - 5.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.99T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 - 6.12T + 67T^{2} \)
71 \( 1 + 1.30T + 71T^{2} \)
73 \( 1 + (-6.64 + 11.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
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

−9.691518096533110532569059514520, −8.225821034042298438140764614560, −7.79016350992430007332786780812, −6.95780702003528688233396844022, −6.17228756478730837870904814485, −5.39960478997760584057941371603, −3.93728792052137146124113706449, −3.19066537601950373741838625337, −2.61767755160859829595112757890, −0.36684251677189532299259891293, 1.01107088250714206820852576281, 2.55353763719748195748115906708, 3.74190725109882921098874425474, 4.71656849220744297603263794224, 5.09287528209116597466582283613, 6.61631911214596341631296740330, 7.03946596747514482140114662050, 8.224068145269719151214738834374, 8.692380817855522753583201806215, 9.668603954456565862736678344635

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