Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.472 - 0.881i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.05 + 2.57i)2-s + (−5.87 − 6.81i)3-s + (2.72 − 15.7i)4-s + (14.3 + 24.7i)5-s + (35.5 + 5.69i)6-s + (22.2 + 12.8i)7-s + (32.3 + 55.2i)8-s + (−11.8 + 80.1i)9-s + (−107. − 38.9i)10-s + (93.9 + 54.2i)11-s + (−123. + 74.1i)12-s + (44.2 + 76.5i)13-s + (−101. + 17.9i)14-s + (84.7 − 243. i)15-s + (−241. − 85.8i)16-s + 504.·17-s + ⋯
L(s)  = 1  + (−0.764 + 0.644i)2-s + (−0.653 − 0.757i)3-s + (0.170 − 0.985i)4-s + (0.572 + 0.991i)5-s + (0.987 + 0.158i)6-s + (0.453 + 0.261i)7-s + (0.504 + 0.863i)8-s + (−0.146 + 0.989i)9-s + (−1.07 − 0.389i)10-s + (0.776 + 0.448i)11-s + (−0.857 + 0.514i)12-s + (0.261 + 0.453i)13-s + (−0.515 + 0.0918i)14-s + (0.376 − 1.08i)15-s + (−0.942 − 0.335i)16-s + 1.74·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.472 - 0.881i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.472 - 0.881i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.792571 + 0.474162i\)
\(L(\frac12)\)  \(\approx\)  \(0.792571 + 0.474162i\)
\(L(3)\)   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 \( 1 + (3.05 - 2.57i)T \)
3 \( 1 + (5.87 + 6.81i)T \)
good5 \( 1 + (-14.3 - 24.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-22.2 - 12.8i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-93.9 - 54.2i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-44.2 - 76.5i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 504.T + 8.35e4T^{2} \)
19 \( 1 + 191. iT - 1.30e5T^{2} \)
23 \( 1 + (831. - 480. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (396. - 687. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (285. - 164. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 209.T + 1.87e6T^{2} \)
41 \( 1 + (528. + 914. i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-2.88e3 - 1.66e3i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (977. + 564. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 1.13e3T + 7.89e6T^{2} \)
59 \( 1 + (4.03e3 - 2.33e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.79e3 + 4.84e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-6.12e3 + 3.53e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 4.43e3iT - 2.54e7T^{2} \)
73 \( 1 + 1.95e3T + 2.83e7T^{2} \)
79 \( 1 + (1.52e3 + 879. i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-2.62e3 - 1.51e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 559.T + 6.27e7T^{2} \)
97 \( 1 + (-1.10e3 + 1.90e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−16.21460446838730375818349239473, −14.60502153074248870759486251150, −13.96367021501745544053366317306, −11.99853735627407385898293204278, −10.86690672374059212253863698253, −9.627509772136765279712867384740, −7.81537263523556973758604003272, −6.69373377942496503668871826136, −5.59423160526479037076211572434, −1.71285057113933243888958110916, 1.02998462060073872800935734061, 3.98386647833208737330135124173, 5.79571485727587099043586727849, 8.144342556566056114024178148409, 9.438152118967103031775694972365, 10.35412367493884210835559678496, 11.69217984692335998453071084228, 12.60736890439762796974808863266, 14.27564789445493142320693563542, 16.15159759148170021960659009316

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