Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.633 + 0.774i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.98 − 5.67i)3-s + (−8.75 − 24.0i)5-s + (−0.632 − 3.58i)7-s + (16.4 − 79.3i)9-s + (−22.6 + 62.1i)11-s + (−236. − 198. i)13-s + (−197. − 118. i)15-s + (205. + 118. i)17-s + (25.2 + 43.7i)19-s + (−24.7 − 21.4i)21-s + (−330. − 58.2i)23-s + (−23.6 + 19.8i)25-s + (−335. − 647. i)27-s + (−402. − 480. i)29-s + (111. − 634. i)31-s + ⋯
L(s)  = 1  + (0.775 − 0.631i)3-s + (−0.350 − 0.962i)5-s + (−0.0129 − 0.0731i)7-s + (0.203 − 0.979i)9-s + (−0.187 + 0.514i)11-s + (−1.39 − 1.17i)13-s + (−0.879 − 0.525i)15-s + (0.710 + 0.410i)17-s + (0.0699 + 0.121i)19-s + (−0.0561 − 0.0486i)21-s + (−0.624 − 0.110i)23-s + (−0.0378 + 0.0317i)25-s + (−0.460 − 0.887i)27-s + (−0.479 − 0.570i)29-s + (0.116 − 0.660i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.633 + 0.774i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.633 + 0.774i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.714858 - 1.50795i\)
\(L(\frac12)\)  \(\approx\)  \(0.714858 - 1.50795i\)
\(L(3)\)   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 \( 1 + (-6.98 + 5.67i)T \)
good5 \( 1 + (8.75 + 24.0i)T + (-478. + 401. i)T^{2} \)
7 \( 1 + (0.632 + 3.58i)T + (-2.25e3 + 821. i)T^{2} \)
11 \( 1 + (22.6 - 62.1i)T + (-1.12e4 - 9.41e3i)T^{2} \)
13 \( 1 + (236. + 198. i)T + (4.95e3 + 2.81e4i)T^{2} \)
17 \( 1 + (-205. - 118. i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-25.2 - 43.7i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (330. + 58.2i)T + (2.62e5 + 9.57e4i)T^{2} \)
29 \( 1 + (402. + 480. i)T + (-1.22e5 + 6.96e5i)T^{2} \)
31 \( 1 + (-111. + 634. i)T + (-8.67e5 - 3.15e5i)T^{2} \)
37 \( 1 + (-978. + 1.69e3i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + (724. - 862. i)T + (-4.90e5 - 2.78e6i)T^{2} \)
43 \( 1 + (-296. - 107. i)T + (2.61e6 + 2.19e6i)T^{2} \)
47 \( 1 + (-3.61e3 + 636. i)T + (4.58e6 - 1.66e6i)T^{2} \)
53 \( 1 - 3.92e3iT - 7.89e6T^{2} \)
59 \( 1 + (-662. - 1.82e3i)T + (-9.28e6 + 7.78e6i)T^{2} \)
61 \( 1 + (-633. - 3.59e3i)T + (-1.30e7 + 4.73e6i)T^{2} \)
67 \( 1 + (1.25e3 + 1.05e3i)T + (3.49e6 + 1.98e7i)T^{2} \)
71 \( 1 + (-6.35e3 - 3.66e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (832. + 1.44e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-8.24e3 + 6.91e3i)T + (6.76e6 - 3.83e7i)T^{2} \)
83 \( 1 + (-4.03e3 - 4.81e3i)T + (-8.24e6 + 4.67e7i)T^{2} \)
89 \( 1 + (-4.49e3 + 2.59e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (3.95e3 + 1.43e3i)T + (6.78e7 + 5.69e7i)T^{2} \)
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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

−12.48553792494059883523973010806, −12.16418853005747302641354789427, −10.24256583020435742550357055459, −9.249365890628324825022835660417, −8.009805942009743381462591722289, −7.45233845069101052754668689375, −5.64084083649715445378546596426, −4.14518909375840185894002766390, −2.44618299261788929828228483619, −0.66946191829003447194433771929, 2.43614296463761110634725644064, 3.61000530557606491523955214732, 5.04845899238727255795428028080, 6.90050472698619165815896353794, 7.84809155520118547291508319470, 9.187943868979785261782823636778, 10.09642200095862859935705709567, 11.12637546050126263662462185034, 12.19916546098295849893524550876, 13.78820201374816343182871273658

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