Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.367 + 0.930i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.85 + 3.54i)2-s + (6.53 − 6.18i)3-s + (−9.11 − 13.1i)4-s + (−14.8 − 25.7i)5-s + (9.78 + 34.6i)6-s + (−51.8 − 29.9i)7-s + (63.5 − 7.86i)8-s + (4.46 − 80.8i)9-s + (118. − 4.89i)10-s + (195. + 112. i)11-s + (−140. − 29.6i)12-s + (−85.8 − 148. i)13-s + (202. − 128. i)14-s + (−256. − 76.4i)15-s + (−90.0 + 239. i)16-s − 99.0·17-s + ⋯
L(s)  = 1  + (−0.464 + 0.885i)2-s + (0.726 − 0.687i)3-s + (−0.569 − 0.822i)4-s + (−0.595 − 1.03i)5-s + (0.271 + 0.962i)6-s + (−1.05 − 0.611i)7-s + (0.992 − 0.122i)8-s + (0.0551 − 0.998i)9-s + (1.18 − 0.0489i)10-s + (1.61 + 0.931i)11-s + (−0.978 − 0.205i)12-s + (−0.508 − 0.880i)13-s + (1.03 − 0.654i)14-s + (−1.14 − 0.339i)15-s + (−0.351 + 0.936i)16-s − 0.342·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.367 + 0.930i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.367 + 0.930i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.864686 - 0.588208i\)
\(L(\frac12)\)  \(\approx\)  \(0.864686 - 0.588208i\)
\(L(3)\)   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 \( 1 + (1.85 - 3.54i)T \)
3 \( 1 + (-6.53 + 6.18i)T \)
good5 \( 1 + (14.8 + 25.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (51.8 + 29.9i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-195. - 112. i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (85.8 + 148. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 99.0T + 8.35e4T^{2} \)
19 \( 1 - 169. iT - 1.30e5T^{2} \)
23 \( 1 + (-310. + 179. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-9.01 + 15.6i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-671. + 387. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 609.T + 1.87e6T^{2} \)
41 \( 1 + (-206. - 357. i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-265. - 153. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.27e3 - 1.31e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 2.03e3T + 7.89e6T^{2} \)
59 \( 1 + (2.25e3 - 1.29e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-708. + 1.22e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-5.19e3 + 2.99e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 1.23e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.06e3T + 2.83e7T^{2} \)
79 \( 1 + (-1.63e3 - 944. i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (5.83e3 + 3.36e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 9.43e3T + 6.27e7T^{2} \)
97 \( 1 + (-7.29e3 + 1.26e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−15.49623922691047156298889167997, −14.47658795166194130429708762088, −13.15417963410401676729536895453, −12.27850752099664731843469804111, −9.821345880967125017888451703198, −8.893363793434215645112656367449, −7.60501221668009106456171740614, −6.51805446023412740443483657638, −4.19594542104458365811755036989, −0.837272819582801971131818610122, 2.83555144083086472289177694763, 3.91701965788238434163069170381, 6.93292878584391085596607715054, 8.795417456927492524264132321481, 9.539632247472633808776290922927, 10.96087158943875622571537259980, 11.91546085284032794969508992606, 13.60724006216600354494516606991, 14.67182885742483944112756706785, 15.95611778200246311700713875750

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