Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.876 - 0.481i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−27.4 + 15.8i)5-s + (37.6 − 65.2i)7-s + (123. + 71.1i)11-s + (96.3 + 166. i)13-s + 325. i·17-s + 314.·19-s + (443. − 256. i)23-s + (188. − 326. i)25-s + (136. + 78.8i)29-s + (183. + 318. i)31-s + 2.38e3i·35-s + 1.73e3·37-s + (342. − 197. i)41-s + (−360. + 624. i)43-s + (−2.15e3 − 1.24e3i)47-s + ⋯
L(s)  = 1  + (−1.09 + 0.633i)5-s + (0.769 − 1.33i)7-s + (1.01 + 0.588i)11-s + (0.569 + 0.987i)13-s + 1.12i·17-s + 0.870·19-s + (0.839 − 0.484i)23-s + (0.301 − 0.522i)25-s + (0.162 + 0.0937i)29-s + (0.191 + 0.331i)31-s + 1.94i·35-s + 1.26·37-s + (0.203 − 0.117i)41-s + (−0.194 + 0.337i)43-s + (−0.974 − 0.562i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.876 - 0.481i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (89, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.876 - 0.481i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.59213 + 0.408660i\)
\(L(\frac12)\)  \(\approx\)  \(1.59213 + 0.408660i\)
\(L(3)\)   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 \( 1 \)
good5 \( 1 + (27.4 - 15.8i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (-37.6 + 65.2i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-123. - 71.1i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-96.3 - 166. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 325. iT - 8.35e4T^{2} \)
19 \( 1 - 314.T + 1.30e5T^{2} \)
23 \( 1 + (-443. + 256. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-136. - 78.8i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-183. - 318. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 1.73e3T + 1.87e6T^{2} \)
41 \( 1 + (-342. + 197. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (360. - 624. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (2.15e3 + 1.24e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 3.98e3iT - 7.89e6T^{2} \)
59 \( 1 + (-2.13e3 + 1.23e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.24e3 - 2.14e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.29e3 - 5.71e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 5.82e3iT - 2.54e7T^{2} \)
73 \( 1 + 8.79e3T + 2.83e7T^{2} \)
79 \( 1 + (-1.93e3 + 3.35e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (1.04e4 + 6.01e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 7.63e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.45e3 + 2.52e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−13.12589645559710564462067820068, −11.64179141015985188637800661001, −11.21602449316188236608783509625, −10.06148279589825324926246049246, −8.535448453231649719435685742854, −7.40781396727476557778019928321, −6.66468366119843418075814248433, −4.45089109264747375366187411470, −3.69375349289803353284616410463, −1.28004882657192918321811050719, 0.949928618562229170525563108687, 3.13900122944042677893112737325, 4.72265259848176459687293811234, 5.84777766530249678316749024603, 7.63806481642167766905892083409, 8.530513325829332297129357384938, 9.353872666618889105825153295056, 11.34515158248514137139066377267, 11.65815535474157293868852627574, 12.67623836926280670128742653794

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