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Properties

Label 2-55-5.4-c3-0-5
Degree $2$
Conductor $55$
Sign $0.597 - 0.801i$
Analytic cond. $3.24510$
Root an. cond. $1.80141$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.245i·2-s + 2.52i·3-s + 7.93·4-s + (−6.68 + 8.96i)5-s + 0.621·6-s + 10.5i·7-s − 3.91i·8-s + 20.6·9-s + (2.20 + 1.64i)10-s + 11·11-s + 20.0i·12-s + 63.0i·13-s + 2.58·14-s + (−22.6 − 16.8i)15-s + 62.5·16-s − 133. i·17-s + ⋯
L(s)  = 1  − 0.0869i·2-s + 0.486i·3-s + 0.992·4-s + (−0.597 + 0.801i)5-s + 0.0422·6-s + 0.567i·7-s − 0.173i·8-s + 0.763·9-s + (0.0697 + 0.0519i)10-s + 0.301·11-s + 0.482i·12-s + 1.34i·13-s + 0.0492·14-s + (−0.389 − 0.290i)15-s + 0.977·16-s − 1.91i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.597 - 0.801i$
Analytic conductor: \(3.24510\)
Root analytic conductor: \(1.80141\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :3/2),\ 0.597 - 0.801i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.41583 + 0.710637i\)
\(L(\frac12)\) \(\approx\) \(1.41583 + 0.710637i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (6.68 - 8.96i)T \)
11 \( 1 - 11T \)
good2 \( 1 + 0.245iT - 8T^{2} \)
3 \( 1 - 2.52iT - 27T^{2} \)
7 \( 1 - 10.5iT - 343T^{2} \)
13 \( 1 - 63.0iT - 2.19e3T^{2} \)
17 \( 1 + 133. iT - 4.91e3T^{2} \)
19 \( 1 + 76.1T + 6.85e3T^{2} \)
23 \( 1 + 169. iT - 1.21e4T^{2} \)
29 \( 1 + 202.T + 2.43e4T^{2} \)
31 \( 1 - 191.T + 2.97e4T^{2} \)
37 \( 1 - 21.5iT - 5.06e4T^{2} \)
41 \( 1 - 305.T + 6.89e4T^{2} \)
43 \( 1 + 285. iT - 7.95e4T^{2} \)
47 \( 1 - 123. iT - 1.03e5T^{2} \)
53 \( 1 - 480. iT - 1.48e5T^{2} \)
59 \( 1 + 364.T + 2.05e5T^{2} \)
61 \( 1 - 9.11T + 2.26e5T^{2} \)
67 \( 1 + 568. iT - 3.00e5T^{2} \)
71 \( 1 + 157.T + 3.57e5T^{2} \)
73 \( 1 - 212. iT - 3.89e5T^{2} \)
79 \( 1 + 792.T + 4.93e5T^{2} \)
83 \( 1 + 587. iT - 5.71e5T^{2} \)
89 \( 1 + 698.T + 7.04e5T^{2} \)
97 \( 1 - 1.83e3iT - 9.12e5T^{2} \)
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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

−15.17170670624021418912404253981, −14.18280251066818275716459794749, −12.31136285392691955326083029534, −11.48104689896880138243527215911, −10.52539903083314285894660361750, −9.200085189964228357109866776729, −7.36455294690950415527808886130, −6.47966976713271647140241072471, −4.29534180238656807250389115103, −2.51496888048124476394422691739, 1.41068921700332330229546635928, 3.89167045678883275687492191662, 5.93538330656280125107810786661, 7.39717005479031148221783000671, 8.187263697860861615547186794245, 10.14329599760203456992339344367, 11.25759200325413225447026582266, 12.57937617922278048576404192599, 13.08025313502288710352283941419, 15.01514863537872588164457238477

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