Properties

Label 1-104-104.51-r1-0-0
Degree $1$
Conductor $104$
Sign $1$
Analytic cond. $11.1763$
Root an. cond. $11.1763$
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  + 3-s + 5-s + 7-s + 9-s − 11-s + 15-s + 17-s − 19-s + 21-s − 23-s + 25-s + 27-s − 29-s + 31-s − 33-s + 35-s + 37-s − 41-s + 43-s + 45-s + 47-s + 49-s + 51-s − 53-s − 55-s − 57-s − 59-s + ⋯
L(s)  = 1  + 3-s + 5-s + 7-s + 9-s − 11-s + 15-s + 17-s − 19-s + 21-s − 23-s + 25-s + 27-s − 29-s + 31-s − 33-s + 35-s + 37-s − 41-s + 43-s + 45-s + 47-s + 49-s + 51-s − 53-s − 55-s − 57-s − 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $1$
Analytic conductor: \(11.1763\)
Root analytic conductor: \(11.1763\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{104} (51, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 104,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.026961367\)
\(L(\frac12)\) \(\approx\) \(3.026961367\)
\(L(1)\) \(\approx\) \(1.848351028\)
\(L(1)\) \(\approx\) \(1.848351028\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 - T \)
5 \( 1 - T \)
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

−29.84701989611595979887838415498, −28.44564835155039314267509579535, −27.34683579446599695703274000034, −26.179035426298324744753156035, −25.5136042256090969234284742691, −24.49165996135023118833477980327, −23.58106383718192372947967658110, −21.803411450417782748075892772944, −21.041348030693650768374336146779, −20.41963646556510260687141719149, −18.86123399616252282130699874817, −18.093458630888906728726652285, −16.88702495720588850573967598507, −15.38870157540188232768568885581, −14.39948108547307587363533898108, −13.60803938070336577775910494778, −12.498785326255633041266279140494, −10.70081274577193705310219431143, −9.76060647617053544131734412883, −8.46879361640387085847337206627, −7.56397652238752042157472116127, −5.827336206221570025657218411822, −4.4680506144893698448488608055, −2.70191826042558408591819428205, −1.62049583024508401057869089681, 1.62049583024508401057869089681, 2.70191826042558408591819428205, 4.4680506144893698448488608055, 5.827336206221570025657218411822, 7.56397652238752042157472116127, 8.46879361640387085847337206627, 9.76060647617053544131734412883, 10.70081274577193705310219431143, 12.498785326255633041266279140494, 13.60803938070336577775910494778, 14.39948108547307587363533898108, 15.38870157540188232768568885581, 16.88702495720588850573967598507, 18.093458630888906728726652285, 18.86123399616252282130699874817, 20.41963646556510260687141719149, 21.041348030693650768374336146779, 21.803411450417782748075892772944, 23.58106383718192372947967658110, 24.49165996135023118833477980327, 25.5136042256090969234284742691, 26.179035426298324744753156035, 27.34683579446599695703274000034, 28.44564835155039314267509579535, 29.84701989611595979887838415498

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