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Properties

Label 1-97-97.96-r0-0-0
Degree $1$
Conductor $97$
Sign $1$
Analytic cond. $0.450466$
Root an. cond. $0.450466$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $1$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{97} (96, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.943996192\)
\(L(\frac12)\) \(\approx\) \(1.943996192\)
\(L(1)\) \(\approx\) \(1.893495323\)
\(L(1)\) \(\approx\) \(1.893495323\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
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 \( 1 + T \)
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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

−30.26966459451872039316313201513, −29.55975815890733306187645621718, −28.05239012591340682006811668163, −26.7423449736172101876969842674, −25.80611663339547860278899419772, −24.69360431769096635791079466598, −23.96179064651779391335593418256, −22.59219451092514543203786014080, −21.93422738587998203728503530778, −20.43187054233479097209448063531, −19.61956313941108531457018242411, −19.16307561130216494674426449551, −16.842355872364980851935272471445, −15.66064621525898780938462147556, −15.00704193792237262796158440501, −13.89874205955462555010939888256, −12.73750976794474924787776389717, −11.910298558178122093760965782721, −10.31886818510980438157940710570, −8.86113553943907597146249210103, −7.41458435128604388582444775025, −6.51632663856962226607151954769, −4.39405248323803031496271391218, −3.62811412250984882597424611474, −2.31457864518104887525881513152, 2.31457864518104887525881513152, 3.62811412250984882597424611474, 4.39405248323803031496271391218, 6.51632663856962226607151954769, 7.41458435128604388582444775025, 8.86113553943907597146249210103, 10.31886818510980438157940710570, 11.910298558178122093760965782721, 12.73750976794474924787776389717, 13.89874205955462555010939888256, 15.00704193792237262796158440501, 15.66064621525898780938462147556, 16.842355872364980851935272471445, 19.16307561130216494674426449551, 19.61956313941108531457018242411, 20.43187054233479097209448063531, 21.93422738587998203728503530778, 22.59219451092514543203786014080, 23.96179064651779391335593418256, 24.69360431769096635791079466598, 25.80611663339547860278899419772, 26.7423449736172101876969842674, 28.05239012591340682006811668163, 29.55975815890733306187645621718, 30.26966459451872039316313201513

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