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Properties

Label 1-1205-1205.1002-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.728 - 0.684i$
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 + (−0.258 + 0.965i)3-s + (0.866 − 0.5i)4-s + i·6-s + (−0.793 + 0.608i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.608 − 0.793i)11-s + (0.258 + 0.965i)12-s + (−0.793 − 0.608i)13-s + (−0.608 + 0.793i)14-s + (0.5 − 0.866i)16-s + (−0.923 + 0.382i)17-s + (−0.965 − 0.258i)18-s + (0.793 − 0.608i)19-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 − 0.5i)4-s + i·6-s + (−0.793 + 0.608i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.608 − 0.793i)11-s + (0.258 + 0.965i)12-s + (−0.793 − 0.608i)13-s + (−0.608 + 0.793i)14-s + (0.5 − 0.866i)16-s + (−0.923 + 0.382i)17-s + (−0.965 − 0.258i)18-s + (0.793 − 0.608i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1002, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.825256738 - 0.7232463383i\)
\(L(\frac12)\) \(\approx\) \(1.825256738 - 0.7232463383i\)
\(L(1)\) \(\approx\) \(1.477826019 - 0.06946036145i\)
\(L(1)\) \(\approx\) \(1.477826019 - 0.06946036145i\)

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 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 + (-0.793 + 0.608i)T \)
11 \( 1 + (-0.608 - 0.793i)T \)
13 \( 1 + (-0.793 - 0.608i)T \)
17 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (0.793 - 0.608i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (0.965 - 0.258i)T \)
31 \( 1 + (0.991 + 0.130i)T \)
37 \( 1 + (0.793 - 0.608i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (-0.258 - 0.965i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + (-0.965 + 0.258i)T \)
71 \( 1 + (-0.991 - 0.130i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.991 - 0.130i)T \)
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

−21.35918621140787881843521547692, −20.4276117868538901472763799368, −19.82438109440812971939845523564, −19.15380005030318718864901624237, −18.072954775292791635703975478540, −17.33383506716387369558231765147, −16.56373256683689538082956039041, −15.936892772778521881345344292106, −14.90615837049537888910931792012, −14.10118401994952523895602450032, −13.39829509670440552849511745694, −12.8252116760993185602348045276, −12.14810004359101765307158507635, −11.40326458133369425611086692052, −10.46039477410821034480212473856, −9.48289724082329769937415538064, −8.11370854292861851209497179840, −7.37367900402316818586666976334, −6.77028194009584359699198994054, −6.17974260433973653576310334275, −4.96684486535005485106260968694, −4.43034280884908087647709962826, −2.963839553555116052953496756298, −2.481852896583137292629881894676, −1.1999596484106591898639863626, 0.60735497605777849982903457873, 2.62987748216210900083124762779, 2.88421430702120332099167707908, 3.93475798761154120327523740076, 4.921250389448258996665910408727, 5.53124097273218146635333775550, 6.24117911620204499640792938862, 7.23385905306029142880197418350, 8.5594484615985488916039499664, 9.44775128573080967424210842434, 10.258719090753358290213793547851, 10.91340165193623964666275777806, 11.72334074567084953813051232706, 12.47905933677329674075379772967, 13.32726310020865512135512321203, 14.0388438039991328676040233761, 15.22534186362362628711677294603, 15.46607867428774415800761862495, 16.111556721755581889513470307741, 17.00414012883277775022941599238, 17.93961448150783521798750439763, 19.18265107303540523676585597129, 19.67762608290508149071503840994, 20.50703000691408486607314620071, 21.35106610580973307087683638742

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