Properties

Label 1-265-265.264-r0-0-0
Degree $1$
Conductor $265$
Sign $1$
Analytic cond. $1.23065$
Root an. cond. $1.23065$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 11-s + 12-s − 13-s − 14-s + 16-s − 17-s + 18-s − 19-s − 21-s + 22-s + 23-s + 24-s − 26-s + 27-s − 28-s + 29-s − 31-s + 32-s + 33-s − 34-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 11-s + 12-s − 13-s − 14-s + 16-s − 17-s + 18-s − 19-s − 21-s + 22-s + 23-s + 24-s − 26-s + 27-s − 28-s + 29-s − 31-s + 32-s + 33-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(265\)    =    \(5 \cdot 53\)
Sign: $1$
Analytic conductor: \(1.23065\)
Root analytic conductor: \(1.23065\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{265} (264, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 265,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.889944179\)
\(L(\frac12)\) \(\approx\) \(2.889944179\)
\(L(1)\) \(\approx\) \(2.310877609\)
\(L(1)\) \(\approx\) \(2.310877609\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
53 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 - T \)
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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

−25.49859302525363913597536644587, −24.95427944777878272100329591948, −24.12205047812361842988598261968, −22.99192404088288870508281276188, −22.04818762528253854752897759125, −21.48080614634161866432138828425, −20.20428654989142760618599957216, −19.64584997562598787042617821556, −19.01675638242183528613701202122, −17.233588936181195841656643513283, −16.26384052388874148749762293876, −15.214740122447156583835080562290, −14.65602220874348243306755398551, −13.590819633074030230466139917178, −12.869939468481312788401873189147, −12.02662914711244307678962442280, −10.6147530789331880070422079443, −9.56840406052627709773244741512, −8.527588215972845503425828320806, −6.98193216015370712857292735492, −6.610152174342837171339649607370, −4.88962861348800571629188305399, −3.86300374786747770729659247284, −2.94119823888816685673799007323, −1.86599231935421321051782142788, 1.86599231935421321051782142788, 2.94119823888816685673799007323, 3.86300374786747770729659247284, 4.88962861348800571629188305399, 6.610152174342837171339649607370, 6.98193216015370712857292735492, 8.527588215972845503425828320806, 9.56840406052627709773244741512, 10.6147530789331880070422079443, 12.02662914711244307678962442280, 12.869939468481312788401873189147, 13.590819633074030230466139917178, 14.65602220874348243306755398551, 15.214740122447156583835080562290, 16.26384052388874148749762293876, 17.233588936181195841656643513283, 19.01675638242183528613701202122, 19.64584997562598787042617821556, 20.20428654989142760618599957216, 21.48080614634161866432138828425, 22.04818762528253854752897759125, 22.99192404088288870508281276188, 24.12205047812361842988598261968, 24.95427944777878272100329591948, 25.49859302525363913597536644587

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