Properties

Label 1-15-15.14-r1-0-0
Degree $1$
Conductor $15$
Sign $1$
Analytic cond. $1.61197$
Root an. cond. $1.61197$
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 − 7-s + 8-s − 11-s − 13-s − 14-s + 16-s + 17-s + 19-s − 22-s + 23-s − 26-s − 28-s − 29-s + 31-s + 32-s + 34-s − 37-s + 38-s − 41-s − 43-s − 44-s + 46-s + 47-s + 49-s − 52-s + ⋯
L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 11-s − 13-s − 14-s + 16-s + 17-s + 19-s − 22-s + 23-s − 26-s − 28-s − 29-s + 31-s + 32-s + 34-s − 37-s + 38-s − 41-s − 43-s − 44-s + 46-s + 47-s + 49-s − 52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $1$
Analytic conductor: \(1.61197\)
Root analytic conductor: \(1.61197\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 15,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.847029343\)
\(L(\frac12)\) \(\approx\) \(1.847029343\)
\(L(1)\) \(\approx\) \(1.622311470\)
\(L(1)\) \(\approx\) \(1.622311470\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + T \)
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

−41.99740257490063314886123066149, −41.22823157698679018330044979235, −39.44089794397602895029589030001, −38.757341359970418830354799182704, −36.99584669791703414864511396216, −35.0970556892378917853228489627, −33.78117919887043491359601713057, −32.32338008181344539433639966495, −31.36470623994313228882952612958, −29.69227751983587214188771803184, −28.65701374833144389761432722966, −26.31976601519719112857065676519, −24.90350116904296767595196452334, −23.36799923867493537660362632137, −22.17639768821376108756020485300, −20.66236305141030677303888216364, −19.10349718525469465744305631936, −16.63355841618558638158823442508, −15.240408529039033166877888159850, −13.48938457315531804053322255533, −12.15969853381077081792324381745, −10.133748155228375797259200600348, −7.26763059826680725286077767720, −5.34319209367996891782367161036, −3.05701820980868915287415506904, 3.05701820980868915287415506904, 5.34319209367996891782367161036, 7.26763059826680725286077767720, 10.133748155228375797259200600348, 12.15969853381077081792324381745, 13.48938457315531804053322255533, 15.240408529039033166877888159850, 16.63355841618558638158823442508, 19.10349718525469465744305631936, 20.66236305141030677303888216364, 22.17639768821376108756020485300, 23.36799923867493537660362632137, 24.90350116904296767595196452334, 26.31976601519719112857065676519, 28.65701374833144389761432722966, 29.69227751983587214188771803184, 31.36470623994313228882952612958, 32.32338008181344539433639966495, 33.78117919887043491359601713057, 35.0970556892378917853228489627, 36.99584669791703414864511396216, 38.757341359970418830354799182704, 39.44089794397602895029589030001, 41.22823157698679018330044979235, 41.99740257490063314886123066149

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