Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.928 - 0.372i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.85 − 3.54i)2-s + (−9.11 − 13.1i)4-s + (14.8 + 25.7i)5-s + (−51.8 − 29.9i)7-s + (−63.5 + 7.86i)8-s + (118. − 4.89i)10-s + (−195. − 112. i)11-s + (−85.8 − 148. i)13-s + (−202. + 128. i)14-s + (−90.0 + 239. i)16-s + 99.0·17-s + 169. i·19-s + (203. − 430. i)20-s + (−761. + 482. i)22-s + (−310. + 179. i)23-s + ⋯
L(s)  = 1  + (0.464 − 0.885i)2-s + (−0.569 − 0.822i)4-s + (0.595 + 1.03i)5-s + (−1.05 − 0.611i)7-s + (−0.992 + 0.122i)8-s + (1.18 − 0.0489i)10-s + (−1.61 − 0.931i)11-s + (−0.508 − 0.880i)13-s + (−1.03 + 0.654i)14-s + (−0.351 + 0.936i)16-s + 0.342·17-s + 0.468i·19-s + (0.508 − 1.07i)20-s + (−1.57 + 0.997i)22-s + (−0.587 + 0.339i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.372i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.928 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.928 - 0.372i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.928 - 0.372i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.134143 + 0.694258i\)
\(L(\frac12)\)  \(\approx\)  \(0.134143 + 0.694258i\)
\(L(3)\)   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 \( 1 + (-1.85 + 3.54i)T \)
3 \( 1 \)
good5 \( 1 + (-14.8 - 25.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (51.8 + 29.9i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (195. + 112. i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (85.8 + 148. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 99.0T + 8.35e4T^{2} \)
19 \( 1 - 169. iT - 1.30e5T^{2} \)
23 \( 1 + (310. - 179. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (9.01 - 15.6i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-671. + 387. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 609.T + 1.87e6T^{2} \)
41 \( 1 + (206. + 357. i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-265. - 153. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (2.27e3 + 1.31e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 2.03e3T + 7.89e6T^{2} \)
59 \( 1 + (-2.25e3 + 1.29e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-708. + 1.22e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-5.19e3 + 2.99e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 1.23e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.06e3T + 2.83e7T^{2} \)
79 \( 1 + (-1.63e3 - 944. i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-5.83e3 - 3.36e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 9.43e3T + 6.27e7T^{2} \)
97 \( 1 + (-7.29e3 + 1.26e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−12.62873812538706007991717451103, −11.13594412923587910280050230436, −10.20505233442878945284149631999, −9.917431601028097806674600907838, −7.977020281689785255248826157406, −6.38762050025639420550312761914, −5.40370043616163145173640950184, −3.42330470343126209978437382024, −2.61121561219941986056569473120, −0.25021714386292857569754764273, 2.60894074963938649029885388312, 4.62220748745947842467587554779, 5.49981697668757930772143574578, 6.71334066482228342161132848186, 8.030121021157372178299110242436, 9.222500841688552996805327367286, 9.926823800874568782594063262408, 12.07451401264143290739583330521, 12.82584613543992457803311512090, 13.34751133059505352378462307344

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