Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $-0.869 - 0.493i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.63 + 3.01i)2-s + (−8.65 − 2.47i)3-s + (−2.15 + 15.8i)4-s + (−10.5 + 18.3i)5-s + (−15.3 − 32.5i)6-s + (−38.6 + 22.3i)7-s + (−53.4 + 35.2i)8-s + (68.7 + 42.7i)9-s + (−83.0 + 16.3i)10-s + (58.6 − 33.8i)11-s + (57.7 − 131. i)12-s + (14.5 − 25.2i)13-s + (−168. − 57.7i)14-s + (136. − 132. i)15-s + (−246. − 68.2i)16-s + 402.·17-s + ⋯
L(s)  = 1  + (0.657 + 0.753i)2-s + (−0.961 − 0.274i)3-s + (−0.134 + 0.990i)4-s + (−0.423 + 0.732i)5-s + (−0.425 − 0.904i)6-s + (−0.788 + 0.455i)7-s + (−0.834 + 0.550i)8-s + (0.849 + 0.527i)9-s + (−0.830 + 0.163i)10-s + (0.485 − 0.280i)11-s + (0.401 − 0.915i)12-s + (0.0861 − 0.149i)13-s + (−0.861 − 0.294i)14-s + (0.607 − 0.588i)15-s + (−0.963 − 0.266i)16-s + 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.869 - 0.493i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ -0.869 - 0.493i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.272195 + 1.03068i\)
\(L(\frac12)\)  \(\approx\)  \(0.272195 + 1.03068i\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-2.63 - 3.01i)T \)
3 \( 1 + (8.65 + 2.47i)T \)
good5 \( 1 + (10.5 - 18.3i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (38.6 - 22.3i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-58.6 + 33.8i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-14.5 + 25.2i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 402.T + 8.35e4T^{2} \)
19 \( 1 - 644. iT - 1.30e5T^{2} \)
23 \( 1 + (335. + 193. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (362. + 627. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-1.09e3 - 629. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.40e3T + 1.87e6T^{2} \)
41 \( 1 + (774. - 1.34e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.62e3 - 935. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-3.61e3 + 2.08e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 906.T + 7.89e6T^{2} \)
59 \( 1 + (-3.91e3 - 2.26e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (1.31e3 + 2.27e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-58.7 - 33.8i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 1.31e3iT - 2.54e7T^{2} \)
73 \( 1 - 9.47e3T + 2.83e7T^{2} \)
79 \( 1 + (-3.78e3 + 2.18e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-659. + 381. i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 8.08e3T + 6.27e7T^{2} \)
97 \( 1 + (3.33e3 + 5.77e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.24153819641067455931257447480, −15.12208978289400217479018053410, −13.90967115558090436587443316189, −12.43317765287322031484985571766, −11.81547355306651402398736006706, −10.12820512639981434341097393276, −7.968335463907891955185508264157, −6.62290951581070048888091683767, −5.66444857477559475804204647984, −3.58008410780970666230741379278, 0.70367603191526192561688938739, 3.85201789034042600245905957486, 5.17957607767965929333269275992, 6.75493024986641302164522971148, 9.335447449625738492564135136123, 10.42621134549597732170899824360, 11.77990390024424673507949140877, 12.48767359197289407264790763593, 13.66786117839259718660299494128, 15.33331473749727436814699090717

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