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Properties

Label 1-101-101.100-r0-0-0
Degree $1$
Conductor $101$
Sign $1$
Analytic cond. $0.469042$
Root an. cond. $0.469042$
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 + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s − 15-s + 16-s + 17-s − 18-s + 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s − 15-s + 16-s + 17-s − 18-s + 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $1$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{101} (100, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5442777346\)
\(L(\frac12)\) \(\approx\) \(0.5442777346\)
\(L(1)\) \(\approx\) \(0.5966686680\)
\(L(1)\) \(\approx\) \(0.5966686680\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
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

−29.41237257634018777525463258235, −28.74074069748193582785606567756, −28.20848177137599200332867265355, −26.78695179585501117520404698578, −25.846089064558513672904240953946, −25.01726057206836243706052778465, −23.720220737070347844788051157773, −22.62911088966144446671792527398, −21.357447911113330402595948172774, −20.581138215702908324795478962114, −18.7807827192402116476863945095, −18.360039451422438626222358284444, −17.157780274954055141462994342931, −16.37407286332616936259211297146, −15.48263461075803276701839043973, −13.44543331099180747849602988818, −12.420396615783522070073860994258, −10.989791807333251388727121383561, −10.12413741084536290344987450302, −9.28098016147305560033383054835, −7.50605990952547477844201961950, −6.26951687272234859133900119159, −5.50395450485475334881882691478, −2.999351933413840279224430062353, −1.16426932189043392075392276036, 1.16426932189043392075392276036, 2.999351933413840279224430062353, 5.50395450485475334881882691478, 6.26951687272234859133900119159, 7.50605990952547477844201961950, 9.28098016147305560033383054835, 10.12413741084536290344987450302, 10.989791807333251388727121383561, 12.420396615783522070073860994258, 13.44543331099180747849602988818, 15.48263461075803276701839043973, 16.37407286332616936259211297146, 17.157780274954055141462994342931, 18.360039451422438626222358284444, 18.7807827192402116476863945095, 20.581138215702908324795478962114, 21.357447911113330402595948172774, 22.62911088966144446671792527398, 23.720220737070347844788051157773, 25.01726057206836243706052778465, 25.846089064558513672904240953946, 26.78695179585501117520404698578, 28.20848177137599200332867265355, 28.74074069748193582785606567756, 29.41237257634018777525463258235

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