Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.572 - 0.820i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.15 + 1.83i)2-s + (1.26 − 7.89i)4-s + (−2.08 − 1.20i)5-s + (−2.30 + 1.32i)7-s + (11.7 + 19.3i)8-s + (6.70 − 1.23i)10-s + (24.1 + 41.8i)11-s + (−20.3 + 35.2i)13-s + (2.51 − 7.09i)14-s + (−60.8 − 19.9i)16-s − 36.3i·17-s + 125. i·19-s + (−12.1 + 14.9i)20-s + (−128. − 45.6i)22-s + (−97.0 + 168. i)23-s + ⋯
L(s)  = 1  + (−0.760 + 0.648i)2-s + (0.158 − 0.987i)4-s + (−0.186 − 0.107i)5-s + (−0.124 + 0.0718i)7-s + (0.520 + 0.853i)8-s + (0.211 − 0.0391i)10-s + (0.661 + 1.14i)11-s + (−0.434 + 0.752i)13-s + (0.0480 − 0.135i)14-s + (−0.950 − 0.312i)16-s − 0.518i·17-s + 1.51i·19-s + (−0.135 + 0.167i)20-s + (−1.24 − 0.442i)22-s + (−0.879 + 1.52i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 - 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.572 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.572 - 0.820i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (71, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.572 - 0.820i)\)
\(L(2)\)  \(\approx\)  \(0.363152 + 0.696064i\)
\(L(\frac12)\)  \(\approx\)  \(0.363152 + 0.696064i\)
\(L(\frac{5}{2})\)   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 \( 1 + (2.15 - 1.83i)T \)
3 \( 1 \)
good5 \( 1 + (2.08 + 1.20i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (2.30 - 1.32i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-24.1 - 41.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (20.3 - 35.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 36.3iT - 4.91e3T^{2} \)
19 \( 1 - 125. iT - 6.85e3T^{2} \)
23 \( 1 + (97.0 - 168. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (153. - 88.6i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-152. - 87.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 199.T + 5.06e4T^{2} \)
41 \( 1 + (-201. - 116. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-252. + 145. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (30.9 + 53.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 352. iT - 1.48e5T^{2} \)
59 \( 1 + (-70.7 + 122. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (7.71 + 13.3i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (131. + 76.0i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 28.0T + 3.57e5T^{2} \)
73 \( 1 - 124.T + 3.89e5T^{2} \)
79 \( 1 + (648. - 374. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (174. + 302. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 416. iT - 7.04e5T^{2} \)
97 \( 1 + (752. + 1.30e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.99174024076398414149451901860, −12.34695111769245078335093232232, −11.46381969508884038424836393993, −9.925796806470276529124717273564, −9.429046505480324388681728657385, −7.996716651687453023710876093992, −7.08274279037669391728447848068, −5.84505176221983523842958708435, −4.31591725774872287699473022251, −1.75994662149358056112491320194, 0.56196377356538095600270773961, 2.68872053212857403403455129540, 4.11359188701843380384669584269, 6.19147850354460976682636463439, 7.59416442451687029328190898975, 8.621357596575330413695808892426, 9.652980423134304759091692823951, 10.83009977247811380748649396795, 11.54562502989479586610968064024, 12.70443314543540025181115293749

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