Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $-0.273 - 0.961i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.29 + 3.78i)2-s + (8.65 + 2.47i)3-s + (−12.6 + 9.79i)4-s + (−10.5 + 18.3i)5-s + (1.84 + 35.9i)6-s + (38.6 − 22.3i)7-s + (−53.4 − 35.2i)8-s + (68.7 + 42.7i)9-s + (−83.0 − 16.3i)10-s + (−58.6 + 33.8i)11-s + (−133. + 53.4i)12-s + (14.5 − 25.2i)13-s + (134. + 117. i)14-s + (−136. + 132. i)15-s + (64.2 − 247. i)16-s + 402.·17-s + ⋯
L(s)  = 1  + (0.323 + 0.946i)2-s + (0.961 + 0.274i)3-s + (−0.790 + 0.611i)4-s + (−0.423 + 0.732i)5-s + (0.0512 + 0.998i)6-s + (0.788 − 0.455i)7-s + (−0.834 − 0.550i)8-s + (0.849 + 0.527i)9-s + (−0.830 − 0.163i)10-s + (−0.485 + 0.280i)11-s + (−0.928 + 0.371i)12-s + (0.0861 − 0.149i)13-s + (0.685 + 0.599i)14-s + (−0.607 + 0.588i)15-s + (0.251 − 0.967i)16-s + 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.273 - 0.961i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ -0.273 - 0.961i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.20850 + 1.59997i\)
\(L(\frac12)\)  \(\approx\)  \(1.20850 + 1.59997i\)
\(L(3)\)   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 \( 1 + (-1.29 - 3.78i)T \)
3 \( 1 + (-8.65 - 2.47i)T \)
good5 \( 1 + (10.5 - 18.3i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (-38.6 + 22.3i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (58.6 - 33.8i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-14.5 + 25.2i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 402.T + 8.35e4T^{2} \)
19 \( 1 + 644. iT - 1.30e5T^{2} \)
23 \( 1 + (-335. - 193. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (362. + 627. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (1.09e3 + 629. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.40e3T + 1.87e6T^{2} \)
41 \( 1 + (774. - 1.34e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.62e3 + 935. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (3.61e3 - 2.08e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 906.T + 7.89e6T^{2} \)
59 \( 1 + (3.91e3 + 2.26e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (1.31e3 + 2.27e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (58.7 + 33.8i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 1.31e3iT - 2.54e7T^{2} \)
73 \( 1 - 9.47e3T + 2.83e7T^{2} \)
79 \( 1 + (3.78e3 - 2.18e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (659. - 381. i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 8.08e3T + 6.27e7T^{2} \)
97 \( 1 + (3.33e3 + 5.77e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−15.59084152135967128375075393915, −14.93270413832105748389639716192, −14.05113109769989241220906778280, −12.96063240136252726452283352219, −11.05239585293979761809209966513, −9.447825115675793368309038163843, −7.928064758789934912345320395168, −7.21889537802643990839323233236, −4.90422197998288461950240221271, −3.30536257212566467805231901937, 1.54599612164484243544478582671, 3.52963417812891111630861577868, 5.24092522914259745310755461607, 8.009558271383238306482944431965, 8.896985668839442189354191443980, 10.38897297649905603055178016283, 12.06188196432569795420935562859, 12.71252203194330225869433521855, 14.14805786415851856702025725474, 14.84317899143278418173092507740

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