Properties

Label 2-108-27.13-c1-0-2
Degree $2$
Conductor $108$
Sign $0.838 + 0.545i$
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  + (1.30 − 1.14i)3-s + (−0.761 − 0.639i)5-s + (1.35 + 0.492i)7-s + (0.396 − 2.97i)9-s + (−2.56 + 2.15i)11-s + (−0.337 + 1.91i)13-s + (−1.72 + 0.0362i)15-s + (1.16 − 2.01i)17-s + (3.38 + 5.86i)19-s + (2.32 − 0.901i)21-s + (−8.41 + 3.06i)23-s + (−0.696 − 3.95i)25-s + (−2.87 − 4.32i)27-s + (0.847 + 4.80i)29-s + (−5.81 + 2.11i)31-s + ⋯
L(s)  = 1  + (0.752 − 0.658i)3-s + (−0.340 − 0.285i)5-s + (0.511 + 0.186i)7-s + (0.132 − 0.991i)9-s + (−0.773 + 0.648i)11-s + (−0.0936 + 0.531i)13-s + (−0.444 + 0.00936i)15-s + (0.281 − 0.487i)17-s + (0.776 + 1.34i)19-s + (0.507 − 0.196i)21-s + (−1.75 + 0.638i)23-s + (−0.139 − 0.790i)25-s + (−0.553 − 0.832i)27-s + (0.157 + 0.892i)29-s + (−1.04 + 0.380i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.838 + 0.545i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1/2),\ 0.838 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16294 - 0.344978i\)
\(L(\frac12)\) \(\approx\) \(1.16294 - 0.344978i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 1.14i)T \)
good5 \( 1 + (0.761 + 0.639i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-1.35 - 0.492i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (2.56 - 2.15i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.337 - 1.91i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.16 + 2.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.38 - 5.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.41 - 3.06i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.847 - 4.80i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (5.81 - 2.11i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (0.0829 - 0.143i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.87 + 10.6i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-6.82 + 5.73i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-6.14 - 2.23i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 + (-3.02 - 2.53i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (7.91 + 2.88i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.00383 - 0.0217i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-4.53 + 7.85i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.26 + 3.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.00 + 5.70i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.898 - 5.09i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-1.91 - 3.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.40 - 4.53i)T + (16.8 - 95.5i)T^{2} \)
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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

−13.83258901621926210898624899569, −12.33253453010258350738249318867, −12.04402383381191244047776887125, −10.31404919118047683982170828863, −9.140105418181650584314450461679, −7.983423096356040017579562531040, −7.30976276931837044449447368007, −5.57317879583325711418212384971, −3.87948875983784716088081480211, −2.01684233813988193756287545892, 2.75974797778797820020452839801, 4.17572146636358562765252027995, 5.58760639510411974810927286354, 7.57468562286328620561759570510, 8.240226442706818056956493053688, 9.568085123575681398692937804473, 10.63109700699153775137892383425, 11.43929052171744359803271737782, 13.03294517667520252198758855801, 13.91316647814853541167106279489

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