Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.342 - 0.939i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (2 + 3.46i)5-s + (3 + 1.73i)7-s − 7.99·8-s + (−3.99 + 6.92i)10-s + (−10.5 − 6.06i)11-s + (11 + 19.0i)13-s + 6.92i·14-s + (−8 − 13.8i)16-s + 11·17-s − 15.5i·19-s − 15.9·20-s − 24.2i·22-s + (21 − 12.1i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.400 + 0.692i)5-s + (0.428 + 0.247i)7-s − 0.999·8-s + (−0.399 + 0.692i)10-s + (−0.954 − 0.551i)11-s + (0.846 + 1.46i)13-s + 0.494i·14-s + (−0.5 − 0.866i)16-s + 0.647·17-s − 0.820i·19-s − 0.799·20-s − 1.10i·22-s + (0.913 − 0.527i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.342 - 0.939i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.342 - 0.939i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.981948 + 1.40236i\)
\(L(\frac12)\)  \(\approx\)  \(0.981948 + 1.40236i\)
\(L(2)\)   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 \( 1 + (-1 - 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (-2 - 3.46i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3 - 1.73i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (10.5 + 6.06i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-11 - 19.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 11T + 289T^{2} \)
19 \( 1 + 15.5iT - 361T^{2} \)
23 \( 1 + (-21 + 12.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-17 + 29.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (6 - 3.46i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 16T + 1.36e3T^{2} \)
41 \( 1 + (-6.5 - 11.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-43.5 - 25.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-3 - 1.73i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 52T + 2.80e3T^{2} \)
59 \( 1 + (46.5 - 26.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-8 + 13.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (100.5 - 58.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 25T + 5.32e3T^{2} \)
79 \( 1 + (24 + 13.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-30 - 17.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 2T + 7.92e3T^{2} \)
97 \( 1 + (-21.5 + 37.2i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.93850122987775560337118558998, −13.12478370062573265831788423363, −11.76846080072790180638911620209, −10.75845189929520487363540718787, −9.203298071683234380438208606820, −8.176963678357732761926164500091, −6.88879958809555512549124770057, −5.92883519417016020947571069029, −4.58068332897633590863243645595, −2.84990598171486041913618377563, 1.30496350118634431438247537243, 3.23571630612650386065815094918, 4.93828920587932712708747906170, 5.71396854540461132195584079105, 7.78669486520028863702416882409, 9.026514674388931796367953414332, 10.27570757075348754178528398541, 10.90826749306954118387315446407, 12.42948474150826786960092143337, 12.92291934826692834874083558352

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