Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.210 - 0.977i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.14 − 3.37i)2-s + (−6.53 − 6.18i)3-s + (−6.83 + 14.4i)4-s + (−14.8 + 25.7i)5-s + (−6.90 + 35.3i)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 + (134. − 52.2i)12-s + (−85.8 + 148. i)13-s + (−212. − 111. i)14-s + (256. − 76.4i)15-s + (−162. − 197. i)16-s − 99.0·17-s + ⋯
L(s)  = 1  + (−0.535 − 0.844i)2-s + (−0.726 − 0.687i)3-s + (−0.427 + 0.904i)4-s + (−0.595 + 1.03i)5-s + (−0.191 + 0.981i)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 + (0.931 − 0.363i)12-s + (−0.508 + 0.880i)13-s + (−1.08 − 0.567i)14-s + (1.14 − 0.339i)15-s + (−0.634 − 0.772i)16-s − 0.342·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.210 - 0.977i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.210 - 0.977i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.264141 + 0.213393i\)
\(L(\frac12)\)  \(\approx\)  \(0.264141 + 0.213393i\)
\(L(3)\)   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 \( 1 + (2.14 + 3.37i)T \)
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

−16.31698862429141746647989339258, −14.60123430637965752208984953418, −13.23852361108551562021690755463, −11.96994749765201561802972943520, −11.04100701160937994273025910984, −10.27790018347361894068655125577, −7.86544749923903034242324290354, −7.25099641307209741666470378275, −4.60524064360513576384767645434, −2.14373438311457880157550492481, 0.29976144366264578282958425453, 4.86850778337616235047840568800, 5.54200408474170376519021758618, 7.892962015978304288026080354857, 8.768167093804448669452138018350, 10.37754285057235737321752783609, 11.51468520080857002418811234528, 13.04344002897624967053096525865, 14.87570386442374088441043572354, 15.75069016445697383004361313291

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