Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $-0.869 + 0.494i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.485 + 3.97i)2-s + (−4.25 + 7.93i)3-s + (−15.5 − 3.85i)4-s + (1.01 − 1.75i)5-s + (−29.4 − 20.7i)6-s + (−20.0 + 11.5i)7-s + (22.8 − 59.7i)8-s + (−44.8 − 67.4i)9-s + (6.48 + 4.88i)10-s + (−4.32 + 2.49i)11-s + (96.6 − 106. i)12-s + (−137. + 238. i)13-s + (−36.1 − 85.1i)14-s + (9.63 + 15.5i)15-s + (226. + 119. i)16-s − 266.·17-s + ⋯
L(s)  = 1  + (−0.121 + 0.992i)2-s + (−0.472 + 0.881i)3-s + (−0.970 − 0.241i)4-s + (0.0406 − 0.0703i)5-s + (−0.817 − 0.576i)6-s + (−0.408 + 0.236i)7-s + (0.357 − 0.934i)8-s + (−0.553 − 0.832i)9-s + (0.0648 + 0.0488i)10-s + (−0.0357 + 0.0206i)11-s + (0.671 − 0.741i)12-s + (−0.815 + 1.41i)13-s + (−0.184 − 0.434i)14-s + (0.0428 + 0.0690i)15-s + (0.883 + 0.468i)16-s − 0.920·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.869 + 0.494i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ -0.869 + 0.494i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.155270 - 0.586562i\)
\(L(\frac12)\)  \(\approx\)  \(0.155270 - 0.586562i\)
\(L(3)\)   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 \( 1 + (0.485 - 3.97i)T \)
3 \( 1 + (4.25 - 7.93i)T \)
good5 \( 1 + (-1.01 + 1.75i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (20.0 - 11.5i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (4.32 - 2.49i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (137. - 238. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 266.T + 8.35e4T^{2} \)
19 \( 1 - 367. iT - 1.30e5T^{2} \)
23 \( 1 + (-544. - 314. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (319. + 553. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (1.19e3 + 687. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 1.46e3T + 1.87e6T^{2} \)
41 \( 1 + (-593. + 1.02e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.43e3 - 825. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (307. - 177. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 5.29e3T + 7.89e6T^{2} \)
59 \( 1 + (-5.22e3 - 3.01e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (833. + 1.44e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-1.90e3 - 1.10e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 524. iT - 2.54e7T^{2} \)
73 \( 1 + 1.49e3T + 2.83e7T^{2} \)
79 \( 1 + (4.44e3 - 2.56e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (6.91e3 - 3.99e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 8.86e3T + 6.27e7T^{2} \)
97 \( 1 + (3.40e3 + 5.90e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−16.53409561418647940528397135358, −15.31308201840386513678496957681, −14.49942650161430628443773979399, −12.97842551565618243084847621856, −11.39626650694947304623735831330, −9.753366129727972285568649292465, −8.993403682928630987560555380277, −7.01788310044901543048279387106, −5.61196325836157947127539420424, −4.18074332597132431215943434071, 0.46226052549709142310936958066, 2.68447698106496330426430514727, 5.10703917877158064469555907103, 7.08095254377108187132841259243, 8.646375445234240967629095377349, 10.32814565863412828973480123864, 11.31035036076702679248286107754, 12.74698251556147024659965875907, 13.15191894093423942053206006484, 14.70403574627992184755705436233

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