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Properties

Label 2-555-185.184-c1-0-14
Degree $2$
Conductor $555$
Sign $0.878 - 0.478i$
Analytic cond. $4.43169$
Root an. cond. $2.10515$
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  − 1.39·2-s + i·3-s − 0.0601·4-s + (2.20 − 0.383i)5-s − 1.39i·6-s + 0.344i·7-s + 2.86·8-s − 9-s + (−3.06 + 0.534i)10-s − 1.00·11-s − 0.0601i·12-s + 4.37·13-s − 0.479i·14-s + (0.383 + 2.20i)15-s − 3.87·16-s − 0.428·17-s + ⋯
L(s)  = 1  − 0.984·2-s + 0.577i·3-s − 0.0300·4-s + (0.985 − 0.171i)5-s − 0.568i·6-s + 0.130i·7-s + 1.01·8-s − 0.333·9-s + (−0.970 + 0.169i)10-s − 0.301·11-s − 0.0173i·12-s + 1.21·13-s − 0.128i·14-s + (0.0991 + 0.568i)15-s − 0.969·16-s − 0.103·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.878 - 0.478i$
Analytic conductor: \(4.43169\)
Root analytic conductor: \(2.10515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 555,\ (\ :1/2),\ 0.878 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951705 + 0.242534i\)
\(L(\frac12)\) \(\approx\) \(0.951705 + 0.242534i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (-2.20 + 0.383i)T \)
37 \( 1 + (-1.95 - 5.76i)T \)
good2 \( 1 + 1.39T + 2T^{2} \)
7 \( 1 - 0.344iT - 7T^{2} \)
11 \( 1 + 1.00T + 11T^{2} \)
13 \( 1 - 4.37T + 13T^{2} \)
17 \( 1 + 0.428T + 17T^{2} \)
19 \( 1 + 5.70iT - 19T^{2} \)
23 \( 1 - 2.35T + 23T^{2} \)
29 \( 1 - 7.62iT - 29T^{2} \)
31 \( 1 + 4.55iT - 31T^{2} \)
41 \( 1 - 2.50T + 41T^{2} \)
43 \( 1 - 8.58T + 43T^{2} \)
47 \( 1 - 7.01iT - 47T^{2} \)
53 \( 1 - 7.41iT - 53T^{2} \)
59 \( 1 - 3.94iT - 59T^{2} \)
61 \( 1 - 2.18iT - 61T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 - 1.92T + 71T^{2} \)
73 \( 1 - 9.22iT - 73T^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 + 0.402iT - 83T^{2} \)
89 \( 1 - 7.34iT - 89T^{2} \)
97 \( 1 + 9.66T + 97T^{2} \)
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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

−10.77868143691478676564149459883, −9.784448628437697321620239852288, −9.031638052978733074131268657971, −8.701759033908757923826797798393, −7.47117452654299600530065592930, −6.29663718094820894182513538259, −5.25944915927937511534289300992, −4.32187000192805945349461202263, −2.72838129927025759593260889203, −1.15078603463770683194078342162, 1.07274350476303095704551291012, 2.22194401144388200900824263374, 3.89293256294332681312105055644, 5.41225757782517469173688537305, 6.25699352040107018729299561499, 7.29751993187828132147512089880, 8.206908274720570112196104244302, 8.905602047224962735272243904646, 9.786053319392903105557483137752, 10.52588417878088290462373637512

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