Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.819 + 0.573i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.210 − 1.98i)2-s + (1.64 − 2.51i)3-s + (−3.91 − 0.836i)4-s + (−0.133 − 0.756i)5-s + (−4.64 − 3.79i)6-s + (3.08 − 3.67i)7-s + (−2.48 + 7.60i)8-s + (−3.60 − 8.24i)9-s + (−1.53 + 0.106i)10-s + (−7.79 − 1.37i)11-s + (−8.52 + 8.44i)12-s + (7.52 + 2.73i)13-s + (−6.65 − 6.90i)14-s + (−2.11 − 0.907i)15-s + (14.5 + 6.54i)16-s + (6.81 + 11.7i)17-s + ⋯
L(s)  = 1  + (0.105 − 0.994i)2-s + (0.547 − 0.836i)3-s + (−0.977 − 0.209i)4-s + (−0.0266 − 0.151i)5-s + (−0.774 − 0.632i)6-s + (0.440 − 0.524i)7-s + (−0.310 + 0.950i)8-s + (−0.400 − 0.916i)9-s + (−0.153 + 0.0106i)10-s + (−0.708 − 0.124i)11-s + (−0.710 + 0.703i)12-s + (0.578 + 0.210i)13-s + (−0.475 − 0.492i)14-s + (−0.141 − 0.0604i)15-s + (0.912 + 0.409i)16-s + (0.400 + 0.693i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.819 + 0.573i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.819 + 0.573i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.448409 - 1.42296i\)
\(L(\frac12)\)  \(\approx\)  \(0.448409 - 1.42296i\)
\(L(2)\)   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 \( 1 + (-0.210 + 1.98i)T \)
3 \( 1 + (-1.64 + 2.51i)T \)
good5 \( 1 + (0.133 + 0.756i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (-3.08 + 3.67i)T + (-8.50 - 48.2i)T^{2} \)
11 \( 1 + (7.79 + 1.37i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (-7.52 - 2.73i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (-6.81 - 11.7i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.9 - 6.89i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (22.4 + 26.7i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (-38.1 + 13.8i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (-14.9 - 17.8i)T + (-166. + 946. i)T^{2} \)
37 \( 1 + (1.05 + 1.82i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-14.0 - 5.09i)T + (1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (-57.3 - 10.1i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (49.8 - 59.3i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 + 43.6T + 2.80e3T^{2} \)
59 \( 1 + (-43.0 + 7.58i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (66.2 + 55.5i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (14.0 - 38.6i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (74.9 - 43.2i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (42.9 - 74.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-7.58 - 20.8i)T + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (7.46 + 20.5i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (66.5 - 115. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-28.3 + 161. i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−12.87134974990047100389976059739, −12.22825706568484247573978621336, −11.00905666847239452121186369592, −10.00693683340143064332173407449, −8.571077224591355805501865510618, −7.902861398253655358871118455640, −6.12799095753135542072679106674, −4.38452332795092080136473002761, −2.84663215105836690021769507617, −1.17448591201863154963332327928, 3.15928375046398102382629307332, 4.74752088392915799996947081310, 5.69169414672368909175504488805, 7.45871806789033478694348964618, 8.380909846154745520214603129508, 9.363330458531622010645339258459, 10.41348141290987720137563874963, 11.83377701605397577428847549034, 13.33871915076596401238447405588, 14.07987175355656895009235439205

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