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Properties

Label 1-1205-1205.1052-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.676 + 0.736i$
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  + (−0.258 + 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.866 − 0.5i)4-s + i·6-s + (−0.130 − 0.991i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.5i)9-s + (0.991 − 0.130i)11-s + (−0.965 − 0.258i)12-s + (−0.130 + 0.991i)13-s + (0.991 + 0.130i)14-s + (0.5 + 0.866i)16-s + (−0.923 + 0.382i)17-s + (0.258 + 0.965i)18-s + (0.130 + 0.991i)19-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.866 − 0.5i)4-s + i·6-s + (−0.130 − 0.991i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.5i)9-s + (0.991 − 0.130i)11-s + (−0.965 − 0.258i)12-s + (−0.130 + 0.991i)13-s + (0.991 + 0.130i)14-s + (0.5 + 0.866i)16-s + (−0.923 + 0.382i)17-s + (0.258 + 0.965i)18-s + (0.130 + 0.991i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1052, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.676 + 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.711731146 + 0.7516265135i\)
\(L(\frac12)\) \(\approx\) \(1.711731146 + 0.7516265135i\)
\(L(1)\) \(\approx\) \(1.225594223 + 0.3816256847i\)
\(L(1)\) \(\approx\) \(1.225594223 + 0.3816256847i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-0.130 - 0.991i)T \)
11 \( 1 + (0.991 - 0.130i)T \)
13 \( 1 + (-0.130 + 0.991i)T \)
17 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (0.130 + 0.991i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (-0.258 + 0.965i)T \)
31 \( 1 + (-0.608 + 0.793i)T \)
37 \( 1 + (0.130 + 0.991i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (0.965 - 0.258i)T \)
59 \( 1 + (0.965 + 0.258i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + (0.258 - 0.965i)T \)
71 \( 1 + (0.608 - 0.793i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.608 - 0.793i)T \)
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

−20.93570737283888157133696915436, −20.21165332892185573899216558989, −19.61280347925313307469619045212, −19.06300185539734305606969452384, −18.1611576190048691899638613075, −17.55577988611517941940398246899, −16.48718223456217513951655414802, −15.434683379920833037818369513221, −14.889694644472730084399097652513, −14.01380512491256019245396916897, −13.00149429733045998287866116107, −12.73406178484792325512252632236, −11.5003672189477042451435708215, −10.97822636809987502592922427815, −9.72395717844733295508916169479, −9.31725544850913835466906171438, −8.683090145556052275161634790277, −7.86392054220880990228237812094, −6.835810554065244905726948380503, −5.44378255734837311086113597216, −4.507216215074221759293439983124, −3.69925934378510563799642527528, −2.59871804488236988183377030356, −2.34618501805735789117881397143, −0.92466307955384522954466032232, 1.07533484345447530841063342173, 1.87992493389154660010560565461, 3.65191566735084513632140942375, 3.960315339363927015725759498975, 5.06505512780816483322118987490, 6.47919306531971036705060524776, 6.922515552958902710164229694572, 7.56897035677202344582124412764, 8.713517350215665454833997401129, 9.03969797107195292750382511477, 9.96911525885592024384940325429, 10.79877514606629305196432459751, 12.10244344429620243626089512850, 13.144931794941493862223843067081, 13.7237190950086994815142208507, 14.41189830625837641756490654848, 14.88633954328898734680454346190, 15.92515020005008101786711052386, 16.69425189066833981011906414610, 17.24819935770315597920193969310, 18.238322323599236574267643657810, 18.99927521589792678823916211117, 19.64155090183939241263107829681, 20.197182364562960820015280330017, 21.285991564892977115970914958538

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