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Properties

Label 2-108-27.7-c1-0-0
Degree $2$
Conductor $108$
Sign $0.673 - 0.739i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.409 + 1.68i)3-s + (−0.103 − 0.584i)5-s + (2.18 + 1.83i)7-s + (−2.66 + 1.37i)9-s + (0.0708 − 0.402i)11-s + (−0.182 − 0.0664i)13-s + (0.942 − 0.413i)15-s + (−3.66 − 6.34i)17-s + (2.06 − 3.57i)19-s + (−2.19 + 4.43i)21-s + (3.12 − 2.61i)23-s + (4.36 − 1.58i)25-s + (−3.41 − 3.92i)27-s + (−9.66 + 3.51i)29-s + (−4.78 + 4.01i)31-s + ⋯
L(s)  = 1  + (0.236 + 0.971i)3-s + (−0.0461 − 0.261i)5-s + (0.826 + 0.693i)7-s + (−0.888 + 0.459i)9-s + (0.0213 − 0.121i)11-s + (−0.0506 − 0.0184i)13-s + (0.243 − 0.106i)15-s + (−0.888 − 1.53i)17-s + (0.473 − 0.820i)19-s + (−0.478 + 0.967i)21-s + (0.650 − 0.546i)23-s + (0.873 − 0.317i)25-s + (−0.656 − 0.754i)27-s + (−1.79 + 0.653i)29-s + (−0.858 + 0.720i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.673 - 0.739i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1/2),\ 0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02169 + 0.451610i\)
\(L(\frac12)\) \(\approx\) \(1.02169 + 0.451610i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.409 - 1.68i)T \)
good5 \( 1 + (0.103 + 0.584i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-2.18 - 1.83i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.0708 + 0.402i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (0.182 + 0.0664i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.66 + 6.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.06 + 3.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.12 + 2.61i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (9.66 - 3.51i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.78 - 4.01i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (2.88 + 4.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.92 - 2.88i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.03 - 5.89i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-8.62 - 7.23i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 3.42T + 53T^{2} \)
59 \( 1 + (-0.813 - 4.61i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-3.34 - 2.80i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (11.4 + 4.15i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-4.09 - 7.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.96 + 3.41i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.47 - 3.08i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (7.91 - 2.88i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-2.38 + 4.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.34 + 7.62i)T + (-91.1 - 33.1i)T^{2} \)
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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

−14.12943716596400342871420472136, −12.82340309138400257850634338086, −11.43131960586901474140538215475, −10.90024053120071231717588539092, −9.214858202316935727827721389643, −8.874123955381132041123299973892, −7.34034944022573230300409239414, −5.41947887710579265353727546284, −4.59158957549343363687647680452, −2.74037447946399299570921014638, 1.78534705225781821451681486265, 3.82557331628205984194117344271, 5.70239433202659064427812759373, 7.08998803499048147374869356603, 7.87893609865003286206145892553, 9.039931427275608585991615975911, 10.66769203058601908538100466200, 11.45903629157848059349817878878, 12.71447718391602746285134954731, 13.50306574731072522491323206339

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