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Properties

Label 2-12e3-9.5-c2-0-8
Degree $2$
Conductor $1728$
Sign $0.445 - 0.895i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.38 − 4.26i)5-s + (1.36 + 2.37i)7-s + (0.932 − 0.538i)11-s + (8.63 − 14.9i)13-s + 17.8i·17-s − 28.8·19-s + (−34.4 − 19.8i)23-s + (23.8 + 41.3i)25-s + (14.6 − 8.44i)29-s + (−12.5 + 21.7i)31-s − 23.3i·35-s − 10.3·37-s + (−33.5 − 19.3i)41-s + (4.54 + 7.87i)43-s + (56.3 − 32.5i)47-s + ⋯
L(s)  = 1  + (−1.47 − 0.853i)5-s + (0.195 + 0.338i)7-s + (0.0847 − 0.0489i)11-s + (0.664 − 1.15i)13-s + 1.05i·17-s − 1.51·19-s + (−1.49 − 0.864i)23-s + (0.955 + 1.65i)25-s + (0.504 − 0.291i)29-s + (−0.405 + 0.702i)31-s − 0.667i·35-s − 0.279·37-s + (−0.817 − 0.472i)41-s + (0.105 + 0.183i)43-s + (1.19 − 0.692i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.445 - 0.895i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.445 - 0.895i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6957132816\)
\(L(\frac12)\) \(\approx\) \(0.6957132816\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (7.38 + 4.26i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-1.36 - 2.37i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.932 + 0.538i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-8.63 + 14.9i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 17.8iT - 289T^{2} \)
19 \( 1 + 28.8T + 361T^{2} \)
23 \( 1 + (34.4 + 19.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-14.6 + 8.44i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (12.5 - 21.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 10.3T + 1.36e3T^{2} \)
41 \( 1 + (33.5 + 19.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-4.54 - 7.87i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-56.3 + 32.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 43.2iT - 2.80e3T^{2} \)
59 \( 1 + (-65.7 - 37.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (2.72 + 4.72i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (22.9 - 39.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 101. iT - 5.04e3T^{2} \)
73 \( 1 - 33.6T + 5.32e3T^{2} \)
79 \( 1 + (2.65 + 4.60i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (16.1 - 9.33i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 49.2iT - 7.92e3T^{2} \)
97 \( 1 + (-85.9 - 148. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−8.835797274085952704459192982664, −8.290430575550413535852100506263, −8.176212871113286016222406205099, −6.95526904273144243513450718423, −5.98792676527318720369636493778, −5.13064390172394789616993749028, −4.06229216682819704955349602290, −3.72502593816771737306536456766, −2.21274859459281935962839030754, −0.810636383387153664019629067972, 0.25382241666334696360486571241, 1.90843268775153629136023905145, 3.14138649919206410892564032126, 4.09822943459875022601646962073, 4.42273850709158249466230600155, 5.95784603330107822109005304601, 6.78839493898834711609347823419, 7.38404739146229332911815318006, 8.103521761801854599275062612439, 8.854546059241765722233675126433

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