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Properties

Label 1-269-269.268-r0-0-0
Degree $1$
Conductor $269$
Sign $1$
Analytic cond. $1.24923$
Root an. cond. $1.24923$
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 + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $1$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{269} (268, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6532026655\)
\(L(\frac12)\) \(\approx\) \(0.6532026655\)
\(L(1)\) \(\approx\) \(0.6218932215\)
\(L(1)\) \(\approx\) \(0.6218932215\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
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 \( 1 + T \)
53 \( 1 + T \)
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

−25.73256509851077808333598143191, −25.09331278876363130871540299106, −24.150827564876309736985913239518, −22.96765993490631218883723753844, −22.05269202187838472675416283903, −21.27341158074738718084357725977, −20.1681164675045031425478844372, −19.042846159830083182018949186738, −18.31957155674643697523846276014, −17.34938957700878474983881355409, −16.80621006128482024851410819147, −16.000420658157678833305364550550, −14.896106057484915076548045127879, −13.25234189768095396903390110372, −12.559994563099513800595414999006, −11.15903158740569814185774963839, −10.66349035453272764713736156500, −9.419849443814690904328178005944, −8.985795246851946041715620106064, −7.08683370976132335859211674498, −6.36722639390807590089416659566, −5.76765533012140547053572147995, −3.92236704292378661257053207381, −2.23953293562152059858703567915, −0.98214167111318163028497643061, 0.98214167111318163028497643061, 2.23953293562152059858703567915, 3.92236704292378661257053207381, 5.76765533012140547053572147995, 6.36722639390807590089416659566, 7.08683370976132335859211674498, 8.985795246851946041715620106064, 9.419849443814690904328178005944, 10.66349035453272764713736156500, 11.15903158740569814185774963839, 12.559994563099513800595414999006, 13.25234189768095396903390110372, 14.896106057484915076548045127879, 16.000420658157678833305364550550, 16.80621006128482024851410819147, 17.34938957700878474983881355409, 18.31957155674643697523846276014, 19.042846159830083182018949186738, 20.1681164675045031425478844372, 21.27341158074738718084357725977, 22.05269202187838472675416283903, 22.96765993490631218883723753844, 24.150827564876309736985913239518, 25.09331278876363130871540299106, 25.73256509851077808333598143191

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