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Properties

Label 1-260-260.3-r0-0-0
Degree $1$
Conductor $260$
Sign $0.971 + 0.235i$
Analytic cond. $1.20743$
Root an. cond. $1.20743$
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)3-s + (0.866 − 0.5i)7-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + 21-s + (0.866 + 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − 31-s + (0.866 − 0.5i)33-s + (0.866 + 0.5i)37-s + (−0.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.866 − 0.5i)7-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + 21-s + (0.866 + 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − 31-s + (0.866 − 0.5i)33-s + (0.866 + 0.5i)37-s + (−0.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.971 + 0.235i$
Analytic conductor: \(1.20743\)
Root analytic conductor: \(1.20743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (0:\ ),\ 0.971 + 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.765813706 + 0.2113404016i\)
\(L(\frac12)\) \(\approx\) \(1.765813706 + 0.2113404016i\)
\(L(1)\) \(\approx\) \(1.476215425 + 0.1374062835i\)
\(L(1)\) \(\approx\) \(1.476215425 + 0.1374062835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 + iT \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + iT \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−25.498700109251755867468120240404, −25.10426857015254291395344355752, −24.21363660695672242423877683054, −23.30298787555012503219915896329, −22.13099652225185990206958922408, −21.051505839833761434760628204219, −20.34898726618220501318564096336, −19.501069042742028809297627188459, −18.35401006214829343301274268908, −17.88271120826290991508972611483, −16.60880951515346421340657472624, −15.11035922426178830840933947789, −14.78340227892520767021431925250, −13.72608729278680841820381798352, −12.62486657504094859814402170828, −11.87267765290051827378669956631, −10.57964475004329991754727035949, −9.21974915111300868477207812514, −8.58781886454260775354200544157, −7.46342147368327215835006319815, −6.58056107745934306520086870577, −5.03663906614722224575279915663, −3.89374106234390505390553479083, −2.43443273398146362921827549267, −1.56770702719305979460957515404, 1.500464515222513451833054181, 2.8688927440393070488999935866, 4.07237614745304444014497652748, 4.90560493889550378368312030095, 6.49426592526206037940687675618, 7.7684849072117759194584494779, 8.596003370966027509426737883581, 9.47010634339518865281956646529, 10.80614494712094477442704092029, 11.339567253735096825713911672512, 13.08523384580037628957232980079, 13.77484696709948139517914806955, 14.74016729261667038352721932385, 15.44215135592472121383385155626, 16.65348561175099027169762563810, 17.45973357422259110956842546431, 18.73866676479374886609010420613, 19.69667252692078543807921311262, 20.31757428670264657533713402195, 21.526528602121779709078332345184, 21.789992568523660838668095610176, 23.32900345077443345986347217085, 24.268005149357403939516108462260, 24.98476679497765309193187512915, 26.05947455871633554169791926348

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