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Properties

Label 1-65-65.64-r0-0-0
Degree $1$
Conductor $65$
Sign $1$
Analytic cond. $0.301858$
Root an. cond. $0.301858$
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 + 14-s + 16-s − 17-s + 18-s − 19-s − 21-s − 22-s − 23-s − 24-s − 27-s + 28-s + 29-s − 31-s + 32-s + 33-s − 34-s + 36-s + 37-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 11-s − 12-s + 14-s + 16-s − 17-s + 18-s − 19-s − 21-s − 22-s − 23-s − 24-s − 27-s + 28-s + 29-s − 31-s + 32-s + 33-s − 34-s + 36-s + 37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.301858\)
Root analytic conductor: \(0.301858\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{65} (64, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.281548412\)
\(L(\frac12)\) \(\approx\) \(1.281548412\)
\(L(1)\) \(\approx\) \(1.377516009\)
\(L(1)\) \(\approx\) \(1.377516009\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
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

−32.22109787572700739652485863195, −31.00404541733588399170993889135, −30.092678002330817092787639559541, −29.016094204356631669967481376467, −28.12888338662632337239938173731, −26.76939867743925485837418514708, −25.16476771219407970981073301924, −23.83956699064194536116258424151, −23.56407838540816564025499102648, −22.06735108878170469214362021812, −21.3697091208261811746449570967, −20.193467803431018690738869498084, −18.41776870615577385339649438562, −17.27939357745346019357160985252, −16.003664427787082911374558503909, −15.012485225098093472528983576032, −13.53090524015025158224887857333, −12.410613901871519922243184528760, −11.25593410876488252680967523872, −10.45037892968077579494035245741, −7.95750219204714275356625722537, −6.53716761270970690712718657039, −5.26148032342419949590911122736, −4.30261251939371171903810676956, −2.048041700432179757689119712004, 2.048041700432179757689119712004, 4.30261251939371171903810676956, 5.26148032342419949590911122736, 6.53716761270970690712718657039, 7.95750219204714275356625722537, 10.45037892968077579494035245741, 11.25593410876488252680967523872, 12.410613901871519922243184528760, 13.53090524015025158224887857333, 15.012485225098093472528983576032, 16.003664427787082911374558503909, 17.27939357745346019357160985252, 18.41776870615577385339649438562, 20.193467803431018690738869498084, 21.3697091208261811746449570967, 22.06735108878170469214362021812, 23.56407838540816564025499102648, 23.83956699064194536116258424151, 25.16476771219407970981073301924, 26.76939867743925485837418514708, 28.12888338662632337239938173731, 29.016094204356631669967481376467, 30.092678002330817092787639559541, 31.00404541733588399170993889135, 32.22109787572700739652485863195

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