Properties

Label 1-260-260.207-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: $\chi_{260} (207, \cdot )$
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

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

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