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Properties

Label 1-97-97.62-r0-0-0
Degree $1$
Conductor $97$
Sign $0.984 + 0.177i$
Analytic cond. $0.450466$
Root an. cond. $0.450466$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + 12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + 12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.984 + 0.177i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ 0.984 + 0.177i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6812112461 + 0.06077788600i\)
\(L(\frac12)\) \(\approx\) \(0.6812112461 + 0.06077788600i\)
\(L(1)\) \(\approx\) \(0.7258287101 + 0.1673577993i\)
\(L(1)\) \(\approx\) \(0.7258287101 + 0.1673577993i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 - T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 - T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (0.5 + 0.866i)T \)
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

−29.998968291784614408834710899214, −28.98073808017353145461681304515, −28.31946471077930245751453026131, −27.23269462502332460257332437426, −25.7618757249928199053150662848, −25.24887530340503977457504530518, −23.579191581228322054612529955220, −22.5746458203421260507480647222, −21.6308558248842729390037666126, −20.609315628749262688496476435486, −18.98346174102802874330042605645, −18.4644643642139953725269429794, −17.79077885702513603290743954532, −16.60250886247709271980756958119, −14.71473166127662646506571147928, −13.48558710941982447477825536253, −12.37191831529327378150089932852, −11.41990343519448639806496367506, −10.4779482662014270278154465421, −9.04780569712659978309230754548, −7.70212202221563816975583607610, −6.50403755988963956287483399531, −4.89039027636142596816274357761, −2.66607715478269998643822191092, −1.80589823295119587976300851754, 0.97191400481347191932559700105, 4.03652831146449899981045911600, 5.27892614609117438367838733884, 6.07485423388910007549660143269, 7.95971642926504135506765707438, 8.90612306447418451130032990768, 10.243530586194669252986884600431, 10.916208768257345192937497608092, 12.961512255750247388676414825227, 14.12038313286421754344575711516, 15.38038758080103741015431366364, 16.37553533679820797422021713522, 17.164371966988131696420888505681, 17.83199541660144958030182198128, 19.53279542790897352529855333410, 20.77847895675605914164139600344, 21.615150722421070496551262907, 23.29606294079251556231126317856, 23.68497301205632680835003344390, 25.04033504094182864279284478993, 26.06917032364464825721711441161, 27.10153588238422129151534148930, 27.811969390934239309738586325, 28.71972301524809232227495891243, 29.80543353039082126094011268761

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