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Properties

Label 2-108-36.23-c5-0-17
Degree $2$
Conductor $108$
Sign $0.996 - 0.0813i$
Analytic cond. $17.3214$
Root an. cond. $4.16190$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.95 + 4.04i)2-s + (−0.713 − 31.9i)4-s + (77.8 + 44.9i)5-s + (124. − 71.6i)7-s + (132. + 123. i)8-s + (−489. + 137. i)10-s + (−278. − 481. i)11-s + (179. − 311. i)13-s + (−200. + 784. i)14-s + (−1.02e3 + 45.6i)16-s − 1.07e3i·17-s − 348. i·19-s + (1.38e3 − 2.52e3i)20-s + (3.04e3 + 780. i)22-s + (1.46e3 − 2.54e3i)23-s + ⋯
L(s)  = 1  + (−0.699 + 0.714i)2-s + (−0.0223 − 0.999i)4-s + (1.39 + 0.803i)5-s + (0.956 − 0.552i)7-s + (0.730 + 0.683i)8-s + (−1.54 + 0.433i)10-s + (−0.693 − 1.20i)11-s + (0.295 − 0.511i)13-s + (−0.274 + 1.07i)14-s + (−0.999 + 0.0445i)16-s − 0.903i·17-s − 0.221i·19-s + (0.772 − 1.40i)20-s + (1.34 + 0.343i)22-s + (0.579 − 1.00i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0813i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.996 - 0.0813i$
Analytic conductor: \(17.3214\)
Root analytic conductor: \(4.16190\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :5/2),\ 0.996 - 0.0813i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.75992 + 0.0717116i\)
\(L(\frac12)\) \(\approx\) \(1.75992 + 0.0717116i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.95 - 4.04i)T \)
3 \( 1 \)
good5 \( 1 + (-77.8 - 44.9i)T + (1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-124. + 71.6i)T + (8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (278. + 481. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-179. + 311. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 1.07e3iT - 1.41e6T^{2} \)
19 \( 1 + 348. iT - 2.47e6T^{2} \)
23 \( 1 + (-1.46e3 + 2.54e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (3.33e3 - 1.92e3i)T + (1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-2.34e3 - 1.35e3i)T + (1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 1.99e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.34e4 - 7.75e3i)T + (5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-1.24e4 + 7.20e3i)T + (7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (1.23e3 + 2.13e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 - 1.33e4iT - 4.18e8T^{2} \)
59 \( 1 + (-1.98e4 + 3.44e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (211. + 366. i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.40e4 + 8.09e3i)T + (6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 6.90e3T + 1.80e9T^{2} \)
73 \( 1 + 1.10e4T + 2.07e9T^{2} \)
79 \( 1 + (6.43e4 - 3.71e4i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-4.85e4 - 8.41e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 7.44e4iT - 5.58e9T^{2} \)
97 \( 1 + (-4.79e4 - 8.30e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−13.29108366062211461278075377345, −10.94997190413515838418631979033, −10.74789544064475819697141926828, −9.512129669852002111343424540989, −8.355415480592742916614260375722, −7.21292559600386321157430566841, −6.05139134722502250367289623008, −5.11024383751042879637530076193, −2.57433311663580420957751323096, −0.933796238663248751213157208023, 1.48226937676738524784292063875, 2.20948710202923098131963166625, 4.49031031806499159362422350878, 5.72234563306014749539855594612, 7.54910267459349732471141258095, 8.730993400932568307233669673408, 9.513505863265079625061171698959, 10.44816804685522352747795682502, 11.65304180724481226515367017442, 12.73334515368172435774187132775

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