Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.734 - 0.678i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0678 − 3.99i)2-s + (−2.72 + 8.57i)3-s + (−15.9 − 0.542i)4-s + (16.6 + 28.7i)5-s + (34.1 + 11.4i)6-s + (39.9 + 23.0i)7-s + (−3.25 + 63.9i)8-s + (−66.0 − 46.8i)9-s + (116. − 64.4i)10-s + (−63.6 − 36.7i)11-s + (48.3 − 135. i)12-s + (151. + 262. i)13-s + (95.0 − 158. i)14-s + (−292. + 63.9i)15-s + (255. + 17.3i)16-s − 182.·17-s + ⋯
L(s)  = 1  + (0.0169 − 0.999i)2-s + (−0.303 + 0.952i)3-s + (−0.999 − 0.0339i)4-s + (0.664 + 1.15i)5-s + (0.947 + 0.319i)6-s + (0.815 + 0.471i)7-s + (−0.0508 + 0.998i)8-s + (−0.816 − 0.578i)9-s + (1.16 − 0.644i)10-s + (−0.525 − 0.303i)11-s + (0.335 − 0.942i)12-s + (0.896 + 1.55i)13-s + (0.484 − 0.807i)14-s + (−1.29 + 0.284i)15-s + (0.997 + 0.0677i)16-s − 0.629·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.734 - 0.678i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.734 - 0.678i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.19293 + 0.466837i\)
\(L(\frac12)\)  \(\approx\)  \(1.19293 + 0.466837i\)
\(L(3)\)   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 \( 1 + (-0.0678 + 3.99i)T \)
3 \( 1 + (2.72 - 8.57i)T \)
good5 \( 1 + (-16.6 - 28.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-39.9 - 23.0i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (63.6 + 36.7i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-151. - 262. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 182.T + 8.35e4T^{2} \)
19 \( 1 + 314. iT - 1.30e5T^{2} \)
23 \( 1 + (290. - 167. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-357. + 618. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-985. + 568. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.00e3T + 1.87e6T^{2} \)
41 \( 1 + (557. + 965. i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-2.18e3 - 1.25e3i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-980. - 566. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 1.05e3T + 7.89e6T^{2} \)
59 \( 1 + (-878. + 507. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (430. - 745. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-559. + 322. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 9.56e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.89e3T + 2.83e7T^{2} \)
79 \( 1 + (6.76e3 + 3.90e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (7.05e3 + 4.07e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 7.65e3T + 6.27e7T^{2} \)
97 \( 1 + (6.36e3 - 1.10e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−15.68440259353847352997467244417, −14.42448992356364265889688938423, −13.65397166174841104275989783851, −11.52625029750906959067172169713, −11.08479776103311377472132780014, −9.871982508350785666652952500776, −8.698392110534094773159176398486, −6.05090264148226486824323572110, −4.37709642359552229900872195901, −2.49368967997991338476370962875, 1.03695869917525662800809481985, 4.95088404849257255972515980759, 6.03307653315911232720596598675, 7.80321504889808466476214772532, 8.568093630748380856491417564852, 10.46688446261001454435505342093, 12.50184295970693700246812225309, 13.24334572411813404676824182801, 14.16643096125612461322267057609, 15.75822099686395140292180257059

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