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Properties

Label 1-115-115.114-r1-0-0
Degree $1$
Conductor $115$
Sign $1$
Analytic cond. $12.3584$
Root an. cond. $12.3584$
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 − 13-s − 14-s + 16-s + 17-s − 18-s − 19-s − 21-s + 22-s + 24-s + 26-s − 27-s + 28-s + 29-s + 31-s − 32-s + 33-s − 34-s + 36-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 11-s − 12-s − 13-s − 14-s + 16-s + 17-s − 18-s − 19-s − 21-s + 22-s + 24-s + 26-s − 27-s + 28-s + 29-s + 31-s − 32-s + 33-s − 34-s + 36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $1$
Analytic conductor: \(12.3584\)
Root analytic conductor: \(12.3584\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{115} (114, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7655757415\)
\(L(\frac12)\) \(\approx\) \(0.7655757415\)
\(L(1)\) \(\approx\) \(0.5859100510\)
\(L(1)\) \(\approx\) \(0.5859100510\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 + T \)
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

−28.91169380460559727924809558606, −27.86333278953377943232532565099, −27.29069743232331526948100366248, −26.27188624375031388934682036097, −24.91086214207107989353364742515, −24.03190057794265756771035757030, −23.177450971703241506796836679085, −21.4586286137345794273495491294, −21.0174448986870693667983213316, −19.449987379882800256832370174957, −18.40707988297470576208407990298, −17.58860297677786602371002052615, −16.8280550359344070343393542505, −15.71949192967577391462731770241, −14.62487790795127624381303314502, −12.64884448372050059139706510891, −11.70765807362370349723209077370, −10.675858537675593398725631568352, −9.89471263947471560470407168315, −8.19888314052421008687589807709, −7.32293223992508400102674481782, −5.91123044908097385471133242660, −4.718166048898683070001685181071, −2.36394811528120522309189224281, −0.796910756948469506963444008487, 0.796910756948469506963444008487, 2.36394811528120522309189224281, 4.718166048898683070001685181071, 5.91123044908097385471133242660, 7.32293223992508400102674481782, 8.19888314052421008687589807709, 9.89471263947471560470407168315, 10.675858537675593398725631568352, 11.70765807362370349723209077370, 12.64884448372050059139706510891, 14.62487790795127624381303314502, 15.71949192967577391462731770241, 16.8280550359344070343393542505, 17.58860297677786602371002052615, 18.40707988297470576208407990298, 19.449987379882800256832370174957, 21.0174448986870693667983213316, 21.4586286137345794273495491294, 23.177450971703241506796836679085, 24.03190057794265756771035757030, 24.91086214207107989353364742515, 26.27188624375031388934682036097, 27.29069743232331526948100366248, 27.86333278953377943232532565099, 28.91169380460559727924809558606

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