Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.798 - 0.602i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.72 − 5.38i)2-s + (−26.0 + 18.5i)4-s + (20.1 − 11.6i)5-s + (−156. − 90.5i)7-s + (145. + 108. i)8-s + (−97.4 − 88.4i)10-s + (−41.2 + 71.4i)11-s + (−70.1 − 121. i)13-s + (−217. + 1.00e3i)14-s + (332. − 968. i)16-s + 901. i·17-s + 2.36e3i·19-s + (−308. + 677. i)20-s + (456. + 98.9i)22-s + (−160. − 277. i)23-s + ⋯
L(s)  = 1  + (−0.305 − 0.952i)2-s + (−0.813 + 0.581i)4-s + (0.360 − 0.208i)5-s + (−1.20 − 0.698i)7-s + (0.801 + 0.597i)8-s + (−0.308 − 0.279i)10-s + (−0.102 + 0.178i)11-s + (−0.115 − 0.199i)13-s + (−0.296 + 1.36i)14-s + (0.324 − 0.945i)16-s + 0.756i·17-s + 1.50i·19-s + (−0.172 + 0.378i)20-s + (0.200 + 0.0435i)22-s + (−0.0631 − 0.109i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.798 - 0.602i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.798 - 0.602i)\)
\(L(3)\)  \(\approx\)  \(0.714512 + 0.239257i\)
\(L(\frac12)\)  \(\approx\)  \(0.714512 + 0.239257i\)
\(L(\frac{7}{2})\)   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 \( 1 + (1.72 + 5.38i)T \)
3 \( 1 \)
good5 \( 1 + (-20.1 + 11.6i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (156. + 90.5i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (41.2 - 71.4i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (70.1 + 121. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 901. iT - 1.41e6T^{2} \)
19 \( 1 - 2.36e3iT - 2.47e6T^{2} \)
23 \( 1 + (160. + 277. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-7.03e3 - 4.06e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-1.53e3 + 885. i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 1.37e4T + 6.93e7T^{2} \)
41 \( 1 + (-4.37e3 + 2.52e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (1.72e4 + 9.96e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (6.62e3 - 1.14e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 2.50e4iT - 4.18e8T^{2} \)
59 \( 1 + (-2.01e4 - 3.49e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (8.93e3 - 1.54e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.90e4 - 1.67e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 5.58e4T + 1.80e9T^{2} \)
73 \( 1 + 6.98e4T + 2.07e9T^{2} \)
79 \( 1 + (3.30e4 + 1.90e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-1.11e4 + 1.92e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 9.89e4iT - 5.58e9T^{2} \)
97 \( 1 + (2.07e4 - 3.59e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−12.84205381937850755435312937916, −11.91636834375438758618337146221, −10.36077471346017187359288178730, −10.04200016207399062323609305896, −8.809578808758137382173294646895, −7.53339523235473342093032363392, −5.98275526721216387698574139638, −4.24065513228417341815169729364, −3.04288842832888975110663085132, −1.29720739591191734296375605526, 0.34577560236413882418469056040, 2.76743512704028931539572202865, 4.74292428488645437860306875927, 6.12584850727906281677425692972, 6.80048897382477978420810004660, 8.286976078830759387662938372875, 9.425248688913022236512421287564, 10.00914589946958039875641658208, 11.61368235149391411251545486279, 13.03375311471187094551837559255

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