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Properties

Label 2-1512-8.5-c1-0-0
Degree $2$
Conductor $1512$
Sign $0.0460 + 0.998i$
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  + (0.725 + 1.21i)2-s + (−0.946 + 1.76i)4-s + 3.06i·5-s − 7-s + (−2.82 + 0.130i)8-s + (−3.72 + 2.22i)10-s − 5.80i·11-s − 3.52i·13-s + (−0.725 − 1.21i)14-s + (−2.20 − 3.33i)16-s − 6.79·17-s + 5.28i·19-s + (−5.40 − 2.90i)20-s + (7.04 − 4.21i)22-s − 5.65·23-s + ⋯
L(s)  = 1  + (0.513 + 0.858i)2-s + (−0.473 + 0.880i)4-s + 1.37i·5-s − 0.377·7-s + (−0.998 + 0.0460i)8-s + (−1.17 + 0.704i)10-s − 1.75i·11-s − 0.977i·13-s + (−0.193 − 0.324i)14-s + (−0.552 − 0.833i)16-s − 1.64·17-s + 1.21i·19-s + (−1.20 − 0.649i)20-s + (1.50 − 0.898i)22-s − 1.17·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0460 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0460 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.0460 + 0.998i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.0460 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1184620128\)
\(L(\frac12)\) \(\approx\) \(0.1184620128\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.725 - 1.21i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 3.06iT - 5T^{2} \)
11 \( 1 + 5.80iT - 11T^{2} \)
13 \( 1 + 3.52iT - 13T^{2} \)
17 \( 1 + 6.79T + 17T^{2} \)
19 \( 1 - 5.28iT - 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 - 1.21iT - 29T^{2} \)
31 \( 1 - 0.107T + 31T^{2} \)
37 \( 1 + 4.90iT - 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 1.85iT - 43T^{2} \)
47 \( 1 - 6.76T + 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + 7.85iT - 59T^{2} \)
61 \( 1 - 12.0iT - 61T^{2} \)
67 \( 1 + 6.66iT - 67T^{2} \)
71 \( 1 - 1.98T + 71T^{2} \)
73 \( 1 + 6.52T + 73T^{2} \)
79 \( 1 - 2.30T + 79T^{2} \)
83 \( 1 - 12.5iT - 83T^{2} \)
89 \( 1 + 7.79T + 89T^{2} \)
97 \( 1 - 4.05T + 97T^{2} \)
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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

−10.21914333617463160974777658906, −8.994226414793295065229619430678, −8.309802189777002174637938767698, −7.56416212253025386828855788185, −6.65054396248127342666597807047, −6.09473993179319413449830938337, −5.50533533797693189746904915808, −4.00348223090335255812618213608, −3.37118442763242289718760431854, −2.55251614592174732078129880797, 0.03656907384532636502011256172, 1.66527094996962878284461031365, 2.35798399638205751348563599018, 3.98323228024692222456455426578, 4.62108387489817739614326136064, 5.02488098819117172964312038582, 6.37400702047442535734435796096, 7.03175251654539553795829989697, 8.466014067847302579604939413661, 9.083402557536998798050122919370

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