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Properties

Label 2-108-4.3-c4-0-1
Degree $2$
Conductor $108$
Sign $-0.909 + 0.414i$
Analytic cond. $11.1639$
Root an. cond. $3.34125$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.36 + 2.16i)2-s + (6.63 + 14.5i)4-s − 43.0·5-s + 14.1i·7-s + (−9.18 + 63.3i)8-s + (−144. − 93.1i)10-s − 209. i·11-s − 195.·13-s + (−30.5 + 47.4i)14-s + (−167. + 193. i)16-s − 303.·17-s + 274. i·19-s + (−285. − 626. i)20-s + (453. − 704. i)22-s + 466. i·23-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (0.414 + 0.909i)4-s − 1.72·5-s + 0.288i·7-s + (−0.143 + 0.989i)8-s + (−1.44 − 0.931i)10-s − 1.73i·11-s − 1.15·13-s + (−0.155 + 0.242i)14-s + (−0.656 + 0.754i)16-s − 1.05·17-s + 0.761i·19-s + (−0.714 − 1.56i)20-s + (0.936 − 1.45i)22-s + 0.881i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.909 + 0.414i$
Analytic conductor: \(11.1639\)
Root analytic conductor: \(3.34125\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :2),\ -0.909 + 0.414i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.104996 - 0.483583i\)
\(L(\frac12)\) \(\approx\) \(0.104996 - 0.483583i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.36 - 2.16i)T \)
3 \( 1 \)
good5 \( 1 + 43.0T + 625T^{2} \)
7 \( 1 - 14.1iT - 2.40e3T^{2} \)
11 \( 1 + 209. iT - 1.46e4T^{2} \)
13 \( 1 + 195.T + 2.85e4T^{2} \)
17 \( 1 + 303.T + 8.35e4T^{2} \)
19 \( 1 - 274. iT - 1.30e5T^{2} \)
23 \( 1 - 466. iT - 2.79e5T^{2} \)
29 \( 1 + 439.T + 7.07e5T^{2} \)
31 \( 1 - 1.22e3iT - 9.23e5T^{2} \)
37 \( 1 - 624.T + 1.87e6T^{2} \)
41 \( 1 + 203.T + 2.82e6T^{2} \)
43 \( 1 - 419. iT - 3.41e6T^{2} \)
47 \( 1 + 1.99e3iT - 4.87e6T^{2} \)
53 \( 1 + 206.T + 7.89e6T^{2} \)
59 \( 1 - 2.32e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.39e3T + 1.38e7T^{2} \)
67 \( 1 + 7.34e3iT - 2.01e7T^{2} \)
71 \( 1 - 8.54e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.24e3T + 2.83e7T^{2} \)
79 \( 1 - 6.15e3iT - 3.89e7T^{2} \)
83 \( 1 + 213. iT - 4.74e7T^{2} \)
89 \( 1 + 8.43e3T + 6.27e7T^{2} \)
97 \( 1 + 1.02e3T + 8.85e7T^{2} \)
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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

−13.66485321247732278917401542076, −12.46030950601739772502375178400, −11.69176705422377020446731570164, −10.96498452387400413376552702938, −8.761015127851082360788251854752, −7.947230808525174524942889841820, −6.93849255350193216306503593680, −5.46620297252736245817391043635, −4.12062893023030389456105580022, −3.07648674690865398081628016939, 0.15863656552820087026755668749, 2.46363680760086784172211193320, 4.20416590340475788526054293121, 4.69250779000836874929325407264, 6.87634026198035650117560459666, 7.58796696938285327833992958565, 9.371194123067764937671281934791, 10.61219473704800953828945429819, 11.60267763859544395137744671758, 12.32336601648193445219447684534

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