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Properties

Label 1-1205-1205.1084-r0-0-0
Degree $1$
Conductor $1205$
Sign $-0.554 + 0.832i$
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  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s − 6-s + (0.258 + 0.965i)7-s + i·8-s + (0.5 − 0.866i)9-s + (−0.258 − 0.965i)11-s + (−0.866 − 0.5i)12-s + (0.258 − 0.965i)13-s + (−0.258 + 0.965i)14-s + (−0.5 + 0.866i)16-s + (0.707 + 0.707i)17-s + (0.866 − 0.5i)18-s + (0.258 + 0.965i)19-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s − 6-s + (0.258 + 0.965i)7-s + i·8-s + (0.5 − 0.866i)9-s + (−0.258 − 0.965i)11-s + (−0.866 − 0.5i)12-s + (0.258 − 0.965i)13-s + (−0.258 + 0.965i)14-s + (−0.5 + 0.866i)16-s + (0.707 + 0.707i)17-s + (0.866 − 0.5i)18-s + (0.258 + 0.965i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ -0.554 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9696816053 + 1.810731060i\)
\(L(\frac12)\) \(\approx\) \(0.9696816053 + 1.810731060i\)
\(L(1)\) \(\approx\) \(1.167998578 + 0.8420249406i\)
\(L(1)\) \(\approx\) \(1.167998578 + 0.8420249406i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.258 + 0.965i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (0.258 - 0.965i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (0.965 - 0.258i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 - iT \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.965 - 0.258i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.965 - 0.258i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−21.18095386633505813090304176057, −20.17154122030064019025947009071, −19.4948923016732914722460376328, −18.71760519049267519028014044492, −17.82769878769057325418551005227, −17.14403292670627669652909976900, −16.201491014907661913050764074288, −15.55664879532803003068157693328, −14.433011131354223530768506974850, −13.66961176221913368811355154618, −13.21816918000614092229674668171, −12.1937122670200836122466315871, −11.65097597929896281650036897035, −10.94122293198973663516674874793, −10.17012465251489783122859511957, −9.40574667090323241608603427955, −7.73602425399381529464761855203, −7.00363878427907234264003815704, −6.5240632317162941193874032028, −5.22520209940159785238597586287, −4.77436926383035360633421782445, −3.92815454723513864734272516516, −2.64399035121538625785703951591, −1.62177799902901917230975883282, −0.781673581184614136125382524228, 1.255361401084884595999978329436, 2.885376471648381446893777215028, 3.44512723416932761559303991366, 4.673100758318903990360522038093, 5.32771322094655746751387093557, 6.01657671728428522982705224183, 6.51996851912009816448524979901, 8.0753035737003229820372723982, 8.350360451219510354820247818455, 9.72581686844870891624202075530, 10.743358375834840056113762775370, 11.31727210897577443797780218190, 12.374265040767521713716500075273, 12.549733520020472020864017356384, 13.752127910397579276912857614624, 14.678947571705771931843071803929, 15.33185794440560635001951332808, 15.92095752881733802019052517785, 16.6941817235681892280690113268, 17.309948041589356449695141206508, 18.27757844794923078362672487286, 18.88605959616305607913347521149, 20.33364208387290952159625217923, 21.04301399025176462787049102219, 21.58144115358497638230205100377

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