Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $-0.781 + 0.624i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.701 − 3.93i)2-s + (5.87 − 6.81i)3-s + (−15.0 + 5.52i)4-s + (14.3 − 24.7i)5-s + (−30.9 − 18.3i)6-s + (−22.2 + 12.8i)7-s + (32.3 + 55.2i)8-s + (−11.8 − 80.1i)9-s + (−107. − 38.9i)10-s + (−93.9 + 54.2i)11-s + (−50.6 + 134. i)12-s + (44.2 − 76.5i)13-s + (66.0 + 78.4i)14-s + (−84.7 − 243. i)15-s + (194. − 165. i)16-s + 504.·17-s + ⋯
L(s)  = 1  + (−0.175 − 0.984i)2-s + (0.653 − 0.757i)3-s + (−0.938 + 0.345i)4-s + (0.572 − 0.991i)5-s + (−0.860 − 0.510i)6-s + (−0.453 + 0.261i)7-s + (0.504 + 0.863i)8-s + (−0.146 − 0.989i)9-s + (−1.07 − 0.389i)10-s + (−0.776 + 0.448i)11-s + (−0.351 + 0.936i)12-s + (0.261 − 0.453i)13-s + (0.337 + 0.400i)14-s + (−0.376 − 1.08i)15-s + (0.761 − 0.648i)16-s + 1.74·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.781 + 0.624i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ -0.781 + 0.624i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.487351 - 1.39028i\)
\(L(\frac12)\)  \(\approx\)  \(0.487351 - 1.39028i\)
\(L(3)\)   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 \( 1 + (0.701 + 3.93i)T \)
3 \( 1 + (-5.87 + 6.81i)T \)
good5 \( 1 + (-14.3 + 24.7i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (22.2 - 12.8i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (93.9 - 54.2i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-44.2 + 76.5i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 504.T + 8.35e4T^{2} \)
19 \( 1 + 191. iT - 1.30e5T^{2} \)
23 \( 1 + (-831. - 480. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (396. + 687. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-285. - 164. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 209.T + 1.87e6T^{2} \)
41 \( 1 + (528. - 914. i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (2.88e3 - 1.66e3i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-977. + 564. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 1.13e3T + 7.89e6T^{2} \)
59 \( 1 + (-4.03e3 - 2.33e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-2.79e3 - 4.84e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (6.12e3 + 3.53e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 4.43e3iT - 2.54e7T^{2} \)
73 \( 1 + 1.95e3T + 2.83e7T^{2} \)
79 \( 1 + (-1.52e3 + 879. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (2.62e3 - 1.51e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 559.T + 6.27e7T^{2} \)
97 \( 1 + (-1.10e3 - 1.90e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−14.96850555312718153887958230327, −13.31575631884624444802961375552, −13.03316311524215318423794150381, −11.88391817706810201273429053173, −9.955700524793639545052090013466, −9.019658477804452117491443009012, −7.76932337294033245656843128194, −5.35234575244163445028834500164, −3.01085044105413750335580619890, −1.18674798133047241269007210662, 3.34048261663206274474119382951, 5.41403129487641309522242133903, 7.00335844480606644805737258690, 8.448567986678863081488282214341, 9.871325958300282976810176757719, 10.58159237083893440814242708759, 13.18681530309036724639941847464, 14.23978359919183391301773586581, 14.86464524163957772580343684955, 16.18022200926068343336527851872

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