Properties

Label 1-260-260.103-r0-0-0
Degree $1$
Conductor $260$
Sign $-0.525 - 0.850i$
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  i·3-s i·7-s − 9-s + 11-s i·17-s − 19-s − 21-s i·23-s + i·27-s − 29-s + 31-s i·33-s + i·37-s − 41-s i·43-s + ⋯
L(s)  = 1  i·3-s i·7-s − 9-s + 11-s i·17-s − 19-s − 21-s i·23-s + i·27-s − 29-s + 31-s i·33-s + i·37-s − 41-s i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(1.20743\)
Root analytic conductor: \(1.20743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (0:\ ),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5240298130 - 0.9399022273i\)
\(L(\frac12)\) \(\approx\) \(0.5240298130 - 0.9399022273i\)
\(L(1)\) \(\approx\) \(0.8616406587 - 0.5224811559i\)
\(L(1)\) \(\approx\) \(0.8616406587 - 0.5224811559i\)

Euler product

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

−26.12929621846673482840100093279, −25.448018277963780800012293684360, −24.51695358134716646883422540093, −23.29604961711813532176148410249, −22.30642871151212083203491110683, −21.64968752953867104387617568091, −20.96795787188346114330921050801, −19.70696314659559692624166104779, −19.07160903930513208954111424024, −17.6163779653943060229814784574, −16.933974322164246615366605956, −15.82894442215341527677355826534, −15.02467993435463580346389611169, −14.38688423745615911680973285522, −12.93578205653392167343305874897, −11.81933847215988694177284791802, −11.02938890969833251148195933042, −9.8205851132503848251079252031, −9.020688203529713328520654731939, −8.17499767018350863586853471399, −6.41497781301966968644366735877, −5.57595180204496535539947848013, −4.34354215904966596823493694016, −3.36225638637384806476546126720, −1.94715944299116212969794872236, 0.7570176309184974908755471599, 2.07237468541800050942093340818, 3.51281542650076306426264501259, 4.79267516270044863300919756209, 6.367193884996188438157390060600, 6.954334627542069342585880372624, 8.04133010630180157634186428427, 9.08647661534909139210807410025, 10.41361938350519121687915808093, 11.48648736940613641189454467354, 12.34300613979200778223145274758, 13.46968338091500937710673918344, 14.0585189311133885839249145279, 15.099246655835675424955827304683, 16.794108008835011527870944476116, 17.04154863274932888547635590474, 18.28707867506311532676228366854, 19.10441076670461995891375158403, 20.04143394157181718712797948990, 20.676670270884443846232540842294, 22.225558355167482614708650742765, 22.95891422544805577329579578987, 23.793536533814473593478531731798, 24.653775720309355157223034856036, 25.41540183248977670991884481713

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