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Properties

Label 1-1205-1205.1038-r0-0-0
Degree $1$
Conductor $1205$
Sign $-0.997 - 0.0694i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.358 − 0.933i)3-s + (0.866 − 0.5i)4-s + (−0.587 − 0.809i)6-s + (0.566 − 0.824i)7-s + (0.707 − 0.707i)8-s + (−0.743 + 0.669i)9-s + (−0.793 + 0.608i)11-s + (−0.777 − 0.629i)12-s + (0.0261 − 0.999i)13-s + (0.333 − 0.942i)14-s + (0.5 − 0.866i)16-s + (−0.760 + 0.649i)17-s + (−0.544 + 0.838i)18-s + (−0.608 − 0.793i)19-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.358 − 0.933i)3-s + (0.866 − 0.5i)4-s + (−0.587 − 0.809i)6-s + (0.566 − 0.824i)7-s + (0.707 − 0.707i)8-s + (−0.743 + 0.669i)9-s + (−0.793 + 0.608i)11-s + (−0.777 − 0.629i)12-s + (0.0261 − 0.999i)13-s + (0.333 − 0.942i)14-s + (0.5 − 0.866i)16-s + (−0.760 + 0.649i)17-s + (−0.544 + 0.838i)18-s + (−0.608 − 0.793i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.997 - 0.0694i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1038, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ -0.997 - 0.0694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06573864919 - 1.891597304i\)
\(L(\frac12)\) \(\approx\) \(0.06573864919 - 1.891597304i\)
\(L(1)\) \(\approx\) \(1.131671508 - 0.9889609038i\)
\(L(1)\) \(\approx\) \(1.131671508 - 0.9889609038i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.358 - 0.933i)T \)
7 \( 1 + (0.566 - 0.824i)T \)
11 \( 1 + (-0.793 + 0.608i)T \)
13 \( 1 + (0.0261 - 0.999i)T \)
17 \( 1 + (-0.760 + 0.649i)T \)
19 \( 1 + (-0.608 - 0.793i)T \)
23 \( 1 + (-0.233 - 0.972i)T \)
29 \( 1 + (-0.0523 + 0.998i)T \)
31 \( 1 + (-0.983 + 0.182i)T \)
37 \( 1 + (-0.0261 - 0.999i)T \)
41 \( 1 + (-0.891 - 0.453i)T \)
43 \( 1 + (0.0784 + 0.996i)T \)
47 \( 1 + (-0.453 - 0.891i)T \)
53 \( 1 + (0.838 + 0.544i)T \)
59 \( 1 + (0.358 - 0.933i)T \)
61 \( 1 + (0.987 + 0.156i)T \)
67 \( 1 + (-0.544 - 0.838i)T \)
71 \( 1 + (-0.902 - 0.430i)T \)
73 \( 1 + (0.852 - 0.522i)T \)
79 \( 1 + (0.987 - 0.156i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (0.130 - 0.991i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.71303341621371745487329742968, −20.9338762802029473277266103206, −20.58883818341334071452107307336, −19.35898145199480124550876799630, −18.407968804668732160256276629065, −17.473638712039264767769709717893, −16.63160280621791086857627557067, −16.00200029221202749798041047801, −15.35096792894392626042158769012, −14.73865252234566093944387298209, −13.89833404435048455750934354596, −13.13438441492444425322251981265, −11.91077180987653759448253233094, −11.567426532718626119011242099751, −10.85880380977117520058306961507, −9.8285576179548305726350676962, −8.76972815036996945405717975817, −8.095877283747675678667587715303, −6.88320326596164293629387360364, −5.89683736486570465429231778632, −5.3959366475675620553960991578, −4.5539568982736533668601034832, −3.80661394465755583843101655608, −2.76620897569085252256963713270, −1.88107135525884288496251922250, 0.50875651664001553450969562746, 1.77511745622391941430289609299, 2.40715639509096904529994574120, 3.58248115477012914626265899914, 4.72504966609641399683152005423, 5.247079962958614583670440750274, 6.31253249393107800605545779135, 7.06929334943576127061505515247, 7.702179700436037818350320741982, 8.64790454052410090627003180745, 10.4069298960094050882723043458, 10.6633625065613101014653493125, 11.43706809473665509935121810521, 12.59666039162574748003578027423, 12.88998695297820452803034896549, 13.56215730589600419268304475263, 14.52147900497744038657928183538, 15.09620521176231645939468784117, 16.136741028245508362496703488080, 17.00519891960306997843764867361, 17.84481773459399743162806754136, 18.37806027461129181165147006350, 19.694724150721299605784454373331, 19.95351056793425733331766099737, 20.728966368724786260374848063573

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