Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.485 - 0.873i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.52 + 3.69i)2-s + (4.76 − 7.63i)3-s + (−11.3 + 11.2i)4-s + (11.0 + 19.1i)5-s + (35.5 + 5.96i)6-s + (82.7 + 47.7i)7-s + (−59.0 − 24.6i)8-s + (−35.5 − 72.7i)9-s + (−54.0 + 70.2i)10-s + (−18.9 − 10.9i)11-s + (32.1 + 140. i)12-s + (−63.1 − 109. i)13-s + (−50.2 + 379. i)14-s + (199. + 6.89i)15-s + (0.961 − 255. i)16-s − 283.·17-s + ⋯
L(s)  = 1  + (0.381 + 0.924i)2-s + (0.529 − 0.848i)3-s + (−0.708 + 0.705i)4-s + (0.442 + 0.767i)5-s + (0.986 + 0.165i)6-s + (1.68 + 0.975i)7-s + (−0.922 − 0.385i)8-s + (−0.438 − 0.898i)9-s + (−0.540 + 0.702i)10-s + (−0.156 − 0.0903i)11-s + (0.223 + 0.974i)12-s + (−0.373 − 0.646i)13-s + (−0.256 + 1.93i)14-s + (0.885 + 0.0306i)15-s + (0.00375 − 0.999i)16-s − 0.982·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.873i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.485 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.485 - 0.873i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.485 - 0.873i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.79465 + 1.05557i\)
\(L(\frac12)\)  \(\approx\)  \(1.79465 + 1.05557i\)
\(L(3)\)   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 \( 1 + (-1.52 - 3.69i)T \)
3 \( 1 + (-4.76 + 7.63i)T \)
good5 \( 1 + (-11.0 - 19.1i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-82.7 - 47.7i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (18.9 + 10.9i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (63.1 + 109. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 283.T + 8.35e4T^{2} \)
19 \( 1 + 323. iT - 1.30e5T^{2} \)
23 \( 1 + (-198. + 114. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (604. - 1.04e3i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-718. + 414. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 318.T + 1.87e6T^{2} \)
41 \( 1 + (164. + 284. i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (179. + 103. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.06e3 + 613. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 2.83e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.27e3 + 737. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-936. + 1.62e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-214. + 123. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 4.30e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.01e3T + 2.83e7T^{2} \)
79 \( 1 + (-6.22e3 - 3.59e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-2.87e3 - 1.66e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 1.54e3T + 6.27e7T^{2} \)
97 \( 1 + (2.91e3 - 5.05e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−15.32444568245852618702497507870, −14.73836921888655025963551358687, −13.85162830213838538158331635689, −12.62648112960925729408072143690, −11.27432323606327890177863859947, −8.952186635593864995146148182425, −7.974303155291473784461376766364, −6.68664915500344077050197863785, −5.17226941174125247687360751797, −2.53768280405582404964005322239, 1.78236657500407336433791857894, 4.23191442437997337788295785348, 5.06361741407098228571245424681, 8.166475788914881573201727375704, 9.395456513372724166119931030880, 10.61096962478871131523462593704, 11.58966865187078553820424523351, 13.35354148106591791088424975648, 14.13657084464396385093454203571, 15.07793469976111346000522661502

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