Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.961 + 0.275i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.619 − 1.90i)2-s + (2.41 + 1.77i)3-s + (−3.23 + 2.35i)4-s + (0.463 + 2.62i)5-s + (1.87 − 5.70i)6-s + (4.34 − 5.18i)7-s + (6.48 + 4.68i)8-s + (2.70 + 8.58i)9-s + (4.70 − 2.50i)10-s + (18.5 + 3.26i)11-s + (−11.9 − 0.0296i)12-s + (−4.76 − 1.73i)13-s + (−12.5 − 5.05i)14-s + (−3.53 + 7.17i)15-s + (4.88 − 15.2i)16-s + (7.37 + 12.7i)17-s + ⋯
L(s)  = 1  + (−0.309 − 0.950i)2-s + (0.806 + 0.591i)3-s + (−0.807 + 0.589i)4-s + (0.0926 + 0.525i)5-s + (0.312 − 0.950i)6-s + (0.621 − 0.740i)7-s + (0.810 + 0.585i)8-s + (0.300 + 0.953i)9-s + (0.470 − 0.250i)10-s + (1.68 + 0.297i)11-s + (−0.999 − 0.00246i)12-s + (−0.366 − 0.133i)13-s + (−0.896 − 0.361i)14-s + (−0.235 + 0.478i)15-s + (0.305 − 0.952i)16-s + (0.433 + 0.751i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.961 + 0.275i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.961 + 0.275i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.50265 - 0.210741i\)
\(L(\frac12)\)  \(\approx\)  \(1.50265 - 0.210741i\)
\(L(2)\)   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 \( 1 + (0.619 + 1.90i)T \)
3 \( 1 + (-2.41 - 1.77i)T \)
good5 \( 1 + (-0.463 - 2.62i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (-4.34 + 5.18i)T + (-8.50 - 48.2i)T^{2} \)
11 \( 1 + (-18.5 - 3.26i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (4.76 + 1.73i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (-7.37 - 12.7i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (28.7 + 16.6i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (11.3 + 13.5i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (38.2 - 13.9i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (-4.43 - 5.28i)T + (-166. + 946. i)T^{2} \)
37 \( 1 + (4.65 + 8.05i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (20.9 + 7.61i)T + (1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (42.0 + 7.40i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-35.0 + 41.7i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 - 25.6T + 2.80e3T^{2} \)
59 \( 1 + (-15.3 + 2.70i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (48.6 + 40.8i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (-11.8 + 32.5i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (-23.7 + 13.6i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-18.2 + 31.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (43.9 + 120. i)T + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (0.482 + 1.32i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (34.1 - 59.2i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (19.7 - 111. i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−13.47858624789024027864058788285, −12.27209422153023238060409141820, −10.94775371715530151392266007707, −10.38415222249828033679380130883, −9.226623634569301027376143791271, −8.336222226783114714643418646110, −7.00658242306179632602365849334, −4.54856446920197323622427909011, −3.65531778803256158436602732827, −1.93318091871290079216383741739, 1.56799814801259263524182695463, 4.10929096054118676590217766270, 5.74898655318855485110352387087, 6.91318161149126622955372496445, 8.171893320870811739210266890295, 8.891485090606603150250708054645, 9.681630899394135580198597484303, 11.66882446540007006209208300906, 12.66728098918641612742867988842, 13.84640210046460504235702216035

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