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Properties

Label 1-1380-1380.1379-r1-0-0
Degree $1$
Conductor $1380$
Sign $1$
Analytic cond. $148.301$
Root an. cond. $148.301$
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  − 7-s − 11-s − 13-s − 17-s + 19-s − 29-s − 31-s + 37-s − 41-s − 43-s − 47-s + 49-s − 53-s + 59-s − 61-s − 67-s + 71-s − 73-s + 77-s + 79-s + 83-s + 89-s + 91-s + 97-s + ⋯
L(s)  = 1  − 7-s − 11-s − 13-s − 17-s + 19-s − 29-s − 31-s + 37-s − 41-s − 43-s − 47-s + 49-s − 53-s + 59-s − 61-s − 67-s + 71-s − 73-s + 77-s + 79-s + 83-s + 89-s + 91-s + 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(148.301\)
Root analytic conductor: \(148.301\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1380} (1379, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5893593421\)
\(L(\frac12)\) \(\approx\) \(0.5893593421\)
\(L(1)\) \(\approx\) \(0.6765506513\)
\(L(1)\) \(\approx\) \(0.6765506513\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
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

−20.3658767966286843996562151290, −20.039221992177374831376767860435, −19.12682146943668816027415468110, −18.39130614082365778364310035925, −17.72888558085601155559500201811, −16.67657192247642188422985939061, −16.18204775092562528950572192483, −15.32680269563277499576915141861, −14.698972758355586226280875857890, −13.4705406993077095135999660368, −13.12872733920025313602173972291, −12.274841096649709166948948942375, −11.38953175481440213806073931569, −10.49981653863182396289497748569, −9.697216746898894561151979386613, −9.16270844171633164030379199792, −7.97636502016261263707209701081, −7.263643997120556503398287279814, −6.46913149547189633061232819628, −5.462850329620598500092959149231, −4.759373526785030874229883191947, −3.55981159820446342446594609262, −2.80230354481385335454061924963, −1.8817374572415559983441180082, −0.3139149186335260245204368167, 0.3139149186335260245204368167, 1.8817374572415559983441180082, 2.80230354481385335454061924963, 3.55981159820446342446594609262, 4.759373526785030874229883191947, 5.462850329620598500092959149231, 6.46913149547189633061232819628, 7.263643997120556503398287279814, 7.97636502016261263707209701081, 9.16270844171633164030379199792, 9.697216746898894561151979386613, 10.49981653863182396289497748569, 11.38953175481440213806073931569, 12.274841096649709166948948942375, 13.12872733920025313602173972291, 13.4705406993077095135999660368, 14.698972758355586226280875857890, 15.32680269563277499576915141861, 16.18204775092562528950572192483, 16.67657192247642188422985939061, 17.72888558085601155559500201811, 18.39130614082365778364310035925, 19.12682146943668816027415468110, 20.039221992177374831376767860435, 20.3658767966286843996562151290

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