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Properties

Label 1-1299-1299.1298-r1-0-0
Degree $1$
Conductor $1299$
Sign $1$
Analytic cond. $139.596$
Root an. cond. $139.596$
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 + 4-s + 5-s − 7-s − 8-s − 10-s − 11-s + 13-s + 14-s + 16-s − 17-s − 19-s + 20-s + 22-s + 23-s + 25-s − 26-s − 28-s + 29-s − 31-s − 32-s + 34-s − 35-s + 37-s + 38-s − 40-s − 41-s + ⋯
L(s)  = 1  − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 11-s + 13-s + 14-s + 16-s − 17-s − 19-s + 20-s + 22-s + 23-s + 25-s − 26-s − 28-s + 29-s − 31-s − 32-s + 34-s − 35-s + 37-s + 38-s − 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1299 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1299 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1299\)    =    \(3 \cdot 433\)
Sign: $1$
Analytic conductor: \(139.596\)
Root analytic conductor: \(139.596\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1299} (1298, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1299,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076997467\)
\(L(\frac12)\) \(\approx\) \(1.076997467\)
\(L(1)\) \(\approx\) \(0.6973250781\)
\(L(1)\) \(\approx\) \(0.6973250781\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
433 \( 1 \)
good2 \( 1 - T \)
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

−20.64523919517486481323575906180, −20.03322215324348454965612977349, −19.067482087016491099153335936063, −18.44365329979995750155082248393, −17.89159749335650592969522154442, −16.98638161202135529721368798186, −16.41886312767295702358659563079, −15.5837148408141104987115538949, −14.99031919563213089842613751065, −13.61654256211972585070692411458, −13.104717280371337538192325093201, −12.3653764446909908131663698917, −10.93874377697094055581276110012, −10.66548139701526550296667259082, −9.791873616722168164379682566800, −9.00427162827141062301462799068, −8.488357855412048743657672260835, −7.287592867975227776432181014988, −6.3972429337378982274351884125, −6.04698539572344314244350341692, −4.84368238407507538501660573897, −3.33076384482182013438637834808, −2.58137424850859032403030944326, −1.71038092560677498166768375671, −0.51927873117231674166319645263, 0.51927873117231674166319645263, 1.71038092560677498166768375671, 2.58137424850859032403030944326, 3.33076384482182013438637834808, 4.84368238407507538501660573897, 6.04698539572344314244350341692, 6.3972429337378982274351884125, 7.287592867975227776432181014988, 8.488357855412048743657672260835, 9.00427162827141062301462799068, 9.791873616722168164379682566800, 10.66548139701526550296667259082, 10.93874377697094055581276110012, 12.3653764446909908131663698917, 13.104717280371337538192325093201, 13.61654256211972585070692411458, 14.99031919563213089842613751065, 15.5837148408141104987115538949, 16.41886312767295702358659563079, 16.98638161202135529721368798186, 17.89159749335650592969522154442, 18.44365329979995750155082248393, 19.067482087016491099153335936063, 20.03322215324348454965612977349, 20.64523919517486481323575906180

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