Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.850 - 0.525i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.61 + 1.70i)2-s + (8.99 − 0.179i)3-s + (10.1 + 12.3i)4-s + (−2.83 − 4.90i)5-s + (32.8 + 14.7i)6-s + (−45.1 − 26.0i)7-s + (15.6 + 62.0i)8-s + (80.9 − 3.23i)9-s + (−1.85 − 22.5i)10-s + (−92.3 − 53.3i)11-s + (93.6 + 109. i)12-s + (61.0 + 105. i)13-s + (−118. − 171. i)14-s + (−26.3 − 43.6i)15-s + (−49.5 + 251. i)16-s − 122.·17-s + ⋯
L(s)  = 1  + (0.904 + 0.427i)2-s + (0.999 − 0.0199i)3-s + (0.634 + 0.772i)4-s + (−0.113 − 0.196i)5-s + (0.912 + 0.409i)6-s + (−0.921 − 0.532i)7-s + (0.244 + 0.969i)8-s + (0.999 − 0.0399i)9-s + (−0.0185 − 0.225i)10-s + (−0.763 − 0.440i)11-s + (0.650 + 0.759i)12-s + (0.361 + 0.625i)13-s + (−0.605 − 0.874i)14-s + (−0.117 − 0.193i)15-s + (−0.193 + 0.981i)16-s − 0.424·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.850 - 0.525i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.850 - 0.525i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(2.66011 + 0.755405i\)
\(L(\frac12)\)  \(\approx\)  \(2.66011 + 0.755405i\)
\(L(3)\)   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 \( 1 + (-3.61 - 1.70i)T \)
3 \( 1 + (-8.99 + 0.179i)T \)
good5 \( 1 + (2.83 + 4.90i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (45.1 + 26.0i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (92.3 + 53.3i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-61.0 - 105. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 122.T + 8.35e4T^{2} \)
19 \( 1 + 593. iT - 1.30e5T^{2} \)
23 \( 1 + (473. - 273. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (367. - 637. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (507. - 292. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 2.28e3T + 1.87e6T^{2} \)
41 \( 1 + (-1.43e3 - 2.48e3i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-1.94e3 - 1.12e3i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-913. - 527. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 4.75e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.86e3 + 1.07e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-33.1 + 57.4i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.55e3 - 2.05e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 5.03e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.70e3T + 2.83e7T^{2} \)
79 \( 1 + (-1.19e3 - 690. i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (2.60e3 + 1.50e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 3.18e3T + 6.27e7T^{2} \)
97 \( 1 + (-2.40e3 + 4.16e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−15.84296278803912112953256366398, −14.47415010067986330713071224732, −13.40502197780587441414850264195, −12.84823536818643042287976194031, −11.02318754620484140040807295640, −9.239188000100904028017334238639, −7.80081401078617700019552170417, −6.52447639988146713233620038309, −4.37507665410733117149713933410, −2.88291986411721686219624858129, 2.43898864845550361743429508548, 3.84495930036503522655964576823, 5.93563230339432418135648919634, 7.66785568702569251287679178420, 9.511543588617210297151581862544, 10.59007931092548581610522343527, 12.45520140431143327669387135248, 13.09373192304370244791911878466, 14.37067809996018723587141266335, 15.36255872078817008790637701970

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