Properties

Label 2-108-108.31-c2-0-22
Degree $2$
Conductor $108$
Sign $0.206 + 0.978i$
Analytic cond. $2.94278$
Root an. cond. $1.71545$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.91 + 0.584i)2-s + (2.24 − 1.98i)3-s + (3.31 − 2.23i)4-s + (0.641 − 3.64i)5-s + (−3.13 + 5.11i)6-s + (−2.33 − 2.78i)7-s + (−5.03 + 6.21i)8-s + (1.08 − 8.93i)9-s + (0.899 + 7.33i)10-s + (−15.6 + 2.76i)11-s + (3.00 − 11.6i)12-s + (13.1 − 4.78i)13-s + (6.09 + 3.96i)14-s + (−5.80 − 9.45i)15-s + (6.00 − 14.8i)16-s + (4.98 − 8.63i)17-s + ⋯
L(s)  = 1  + (−0.956 + 0.292i)2-s + (0.748 − 0.663i)3-s + (0.829 − 0.558i)4-s + (0.128 − 0.728i)5-s + (−0.522 + 0.852i)6-s + (−0.333 − 0.397i)7-s + (−0.629 + 0.776i)8-s + (0.120 − 0.992i)9-s + (0.0899 + 0.733i)10-s + (−1.42 + 0.251i)11-s + (0.250 − 0.968i)12-s + (1.01 − 0.367i)13-s + (0.435 + 0.282i)14-s + (−0.386 − 0.630i)15-s + (0.375 − 0.926i)16-s + (0.293 − 0.507i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.206 + 0.978i$
Analytic conductor: \(2.94278\)
Root analytic conductor: \(1.71545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1),\ 0.206 + 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.816917 - 0.662414i\)
\(L(\frac12)\) \(\approx\) \(0.816917 - 0.662414i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.91 - 0.584i)T \)
3 \( 1 + (-2.24 + 1.98i)T \)
good5 \( 1 + (-0.641 + 3.64i)T + (-23.4 - 8.55i)T^{2} \)
7 \( 1 + (2.33 + 2.78i)T + (-8.50 + 48.2i)T^{2} \)
11 \( 1 + (15.6 - 2.76i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (-13.1 + 4.78i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (-4.98 + 8.63i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-14.6 + 8.46i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (0.374 - 0.446i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (-16.6 - 6.04i)T + (644. + 540. i)T^{2} \)
31 \( 1 + (18.3 - 21.8i)T + (-166. - 946. i)T^{2} \)
37 \( 1 + (31.7 - 55.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-59.2 + 21.5i)T + (1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (-1.70 + 0.300i)T + (1.73e3 - 632. i)T^{2} \)
47 \( 1 + (-51.7 - 61.6i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 + 66.4T + 2.80e3T^{2} \)
59 \( 1 + (-65.4 - 11.5i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (-5.43 + 4.55i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (12.7 + 35.1i)T + (-3.43e3 + 2.88e3i)T^{2} \)
71 \( 1 + (-12.7 - 7.38i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-66.1 - 114. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (24.4 - 67.1i)T + (-4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (-8.25 + 22.6i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (29.6 + 51.3i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-13.3 - 75.5i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.25321633625374570522919488528, −12.35581249114580813252277724557, −10.87062962433356462282949740589, −9.744575379408408148052442651859, −8.742845404815420626935585720174, −7.87377300312316149028435296824, −6.89656580168809937672784815920, −5.39021091300331919864256874087, −2.91661226189439196986513294958, −1.02489927911311185568326817667, 2.43905948194785939601121678850, 3.55675896363506002617186241136, 5.85220198238283464566936416724, 7.45818493396999781870918505430, 8.418410002052440953803658284274, 9.445740053926267141193700837401, 10.47546448535319067316524446209, 11.02208460313660796606563934209, 12.59422631182389534622517977474, 13.73976261662859161222261711027

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