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Properties

Label 1-93-93.92-r0-0-0
Degree $1$
Conductor $93$
Sign $1$
Analytic cond. $0.431890$
Root an. cond. $0.431890$
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 − 32-s − 34-s − 35-s − 37-s − 38-s + 40-s − 41-s − 43-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 − 32-s − 34-s − 35-s − 37-s − 38-s + 40-s − 41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $1$
Analytic conductor: \(0.431890\)
Root analytic conductor: \(0.431890\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{93} (92, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6463882676\)
\(L(\frac12)\) \(\approx\) \(0.6463882676\)
\(L(1)\) \(\approx\) \(0.6980975828\)
\(L(1)\) \(\approx\) \(0.6980975828\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
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 \( 1 + T \)
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

−30.26894661363184977652819679892, −29.13905505248956128918760654205, −27.783752161004323503092243784889, −27.31812329870420989204006237363, −26.521252010624751365321100760965, −24.956377810014836349458855546862, −24.35425241125656561041315713588, −23.13389621357033109853129625168, −21.61437279880648113363784687092, −20.40030825270576800426378814414, −19.56797667291413676224705512310, −18.61641015162260692905051547214, −17.38514862043382879917342212004, −16.53293474971464328222143232570, −15.2183085794300512386783948356, −14.405059431081373406762477900138, −12.06254402016068766189096253464, −11.63856129187017707571303652051, −10.27376283236076307735704265953, −8.906088181404724898552890998775, −7.83426023162166228485446494924, −6.95142207703026649152112335629, −5.00169634180100983262293205910, −3.22956829113678628924654860399, −1.30110184040805982023272325805, 1.30110184040805982023272325805, 3.22956829113678628924654860399, 5.00169634180100983262293205910, 6.95142207703026649152112335629, 7.83426023162166228485446494924, 8.906088181404724898552890998775, 10.27376283236076307735704265953, 11.63856129187017707571303652051, 12.06254402016068766189096253464, 14.405059431081373406762477900138, 15.2183085794300512386783948356, 16.53293474971464328222143232570, 17.38514862043382879917342212004, 18.61641015162260692905051547214, 19.56797667291413676224705512310, 20.40030825270576800426378814414, 21.61437279880648113363784687092, 23.13389621357033109853129625168, 24.35425241125656561041315713588, 24.956377810014836349458855546862, 26.521252010624751365321100760965, 27.31812329870420989204006237363, 27.783752161004323503092243784889, 29.13905505248956128918760654205, 30.26894661363184977652819679892

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