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Properties

Label 2-1512-63.47-c1-0-5
Degree $2$
Conductor $1512$
Sign $-0.311 - 0.950i$
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  + 0.623·5-s + (−0.996 − 2.45i)7-s + 5.19i·11-s + (−2.74 + 1.58i)13-s + (−0.437 − 0.757i)17-s + (−1.41 − 0.819i)19-s + 8.25i·23-s − 4.61·25-s + (−4.96 − 2.86i)29-s + (4.02 + 2.32i)31-s + (−0.621 − 1.52i)35-s + (1.24 − 2.15i)37-s + (3.52 + 6.10i)41-s + (−1.56 + 2.70i)43-s + (4.73 + 8.19i)47-s + ⋯
L(s)  = 1  + 0.278·5-s + (−0.376 − 0.926i)7-s + 1.56i·11-s + (−0.760 + 0.438i)13-s + (−0.106 − 0.183i)17-s + (−0.325 − 0.187i)19-s + 1.72i·23-s − 0.922·25-s + (−0.921 − 0.532i)29-s + (0.722 + 0.417i)31-s + (−0.105 − 0.258i)35-s + (0.204 − 0.353i)37-s + (0.550 + 0.953i)41-s + (−0.238 + 0.413i)43-s + (0.690 + 1.19i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.311 - 0.950i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9453251162\)
\(L(\frac12)\) \(\approx\) \(0.9453251162\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.996 + 2.45i)T \)
good5 \( 1 - 0.623T + 5T^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 + (2.74 - 1.58i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.437 + 0.757i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.41 + 0.819i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.25iT - 23T^{2} \)
29 \( 1 + (4.96 + 2.86i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.02 - 2.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.24 + 2.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.52 - 6.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.56 - 2.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.73 - 8.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.15 + 0.665i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.18 + 5.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.65 + 5.57i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.04 - 10.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (11.6 - 6.73i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.84 + 8.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.192 + 0.332i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.0198 - 0.0344i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.94 - 3.43i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.758608258159179302693128173238, −9.249396127794533880553018737574, −7.81580370317587491568086954271, −7.34217412126097840302950344193, −6.64727649722275101408366050194, −5.58731193015600902349735077886, −4.55420440900656023206384607034, −3.94113017583295379869492152265, −2.58232363127883468248721159322, −1.51776325423668418551748185579, 0.35780870100812110718645693862, 2.18086902975232508692246256516, 2.98298956485511535439319212206, 4.06372543880166588018939734719, 5.33636076680519001879948167520, 5.89075209846226405829118700838, 6.61225882110911244344962030293, 7.76294412558777631707235873383, 8.617497489040474971526763063298, 9.030120503107092388857878503645

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