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Properties

Label 2-1512-63.25-c1-0-14
Degree $2$
Conductor $1512$
Sign $0.477 + 0.878i$
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  + (−0.891 + 1.54i)5-s + (−2.54 − 0.727i)7-s + (−2.80 − 4.86i)11-s + (3.14 + 5.43i)13-s + (−0.646 + 1.11i)17-s + (0.559 + 0.968i)19-s + (3.80 − 6.59i)23-s + (0.909 + 1.57i)25-s + (1.57 − 2.72i)29-s + 1.00·31-s + (3.39 − 3.28i)35-s + (−5.96 − 10.3i)37-s + (−4.14 − 7.17i)41-s + (2.34 − 4.06i)43-s + 1.94·47-s + ⋯
L(s)  = 1  + (−0.398 + 0.690i)5-s + (−0.961 − 0.274i)7-s + (−0.846 − 1.46i)11-s + (0.870 + 1.50i)13-s + (−0.156 + 0.271i)17-s + (0.128 + 0.222i)19-s + (0.794 − 1.37i)23-s + (0.181 + 0.315i)25-s + (0.292 − 0.506i)29-s + 0.180·31-s + (0.573 − 0.554i)35-s + (−0.980 − 1.69i)37-s + (−0.646 − 1.12i)41-s + (0.358 − 0.620i)43-s + 0.283·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.477 + 0.878i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.027491582\)
\(L(\frac12)\) \(\approx\) \(1.027491582\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 + 0.727i)T \)
good5 \( 1 + (0.891 - 1.54i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.80 + 4.86i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.14 - 5.43i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.646 - 1.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.559 - 0.968i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.80 + 6.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.57 + 2.72i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.00T + 31T^{2} \)
37 \( 1 + (5.96 + 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.14 + 7.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.34 + 4.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.94T + 47T^{2} \)
53 \( 1 + (-4.45 + 7.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.38T + 59T^{2} \)
61 \( 1 - 4.82T + 61T^{2} \)
67 \( 1 + 2.55T + 67T^{2} \)
71 \( 1 - 8.86T + 71T^{2} \)
73 \( 1 + (-5.67 + 9.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + (-1.60 + 2.77i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.404 - 0.700i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.10 + 1.91i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.082771481883522805386430211750, −8.742405796360415514278894927418, −7.66759069836195089792147231240, −6.73331976952458675900519591603, −6.33416062651266306507787758409, −5.29063244946650799027142779220, −3.91038776310601607774618935182, −3.41981950767687373259027546717, −2.32644934855566368746139494537, −0.47347518252447484333912654853, 1.10169580777320808425907946834, 2.71955319422691331564386876900, 3.48491187299468892843983255040, 4.78651550679450212836471548952, 5.30042135085476445196272072545, 6.38248839065686792566172381610, 7.28171276302186655192905553038, 8.038160148180668577384027804689, 8.779777900650452296811204549991, 9.684094808766847999967587487395

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