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Properties

Label 1-97-97.20-r1-0-0
Degree $1$
Conductor $97$
Sign $0.0569 + 0.998i$
Analytic cond. $10.4240$
Root an. cond. $10.4240$
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  + (−0.923 + 0.382i)2-s + (−0.382 + 0.923i)3-s + (0.707 − 0.707i)4-s + (−0.195 − 0.980i)5-s − i·6-s + (−0.195 + 0.980i)7-s + (−0.382 + 0.923i)8-s + (−0.707 − 0.707i)9-s + (0.555 + 0.831i)10-s + (0.382 − 0.923i)11-s + (0.382 + 0.923i)12-s + (0.980 − 0.195i)13-s + (−0.195 − 0.980i)14-s + (0.980 + 0.195i)15-s − i·16-s + (0.980 − 0.195i)17-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)2-s + (−0.382 + 0.923i)3-s + (0.707 − 0.707i)4-s + (−0.195 − 0.980i)5-s − i·6-s + (−0.195 + 0.980i)7-s + (−0.382 + 0.923i)8-s + (−0.707 − 0.707i)9-s + (0.555 + 0.831i)10-s + (0.382 − 0.923i)11-s + (0.382 + 0.923i)12-s + (0.980 − 0.195i)13-s + (−0.195 − 0.980i)14-s + (0.980 + 0.195i)15-s − i·16-s + (0.980 − 0.195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0569 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0569 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.0569 + 0.998i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.0569 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6211692305 + 0.5867468437i\)
\(L(\frac12)\) \(\approx\) \(0.6211692305 + 0.5867468437i\)
\(L(1)\) \(\approx\) \(0.6094327547 + 0.2554863616i\)
\(L(1)\) \(\approx\) \(0.6094327547 + 0.2554863616i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.923 + 0.382i)T \)
3 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (-0.195 - 0.980i)T \)
7 \( 1 + (-0.195 + 0.980i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (0.980 - 0.195i)T \)
17 \( 1 + (0.980 - 0.195i)T \)
19 \( 1 + (0.195 + 0.980i)T \)
23 \( 1 + (-0.555 + 0.831i)T \)
29 \( 1 + (-0.555 + 0.831i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (-0.831 + 0.555i)T \)
41 \( 1 + (0.831 + 0.555i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.382 + 0.923i)T \)
59 \( 1 + (0.555 + 0.831i)T \)
61 \( 1 + T \)
67 \( 1 + (0.980 - 0.195i)T \)
71 \( 1 + (0.831 - 0.555i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (0.195 + 0.980i)T \)
89 \( 1 + (0.382 - 0.923i)T \)
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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

−29.86449929226723979050917170059, −28.48153514068571705276653677686, −27.75032807921149559095742547604, −26.20403195262977942684213346305, −25.88902904524132178165730530325, −24.498475693013876318854458121021, −23.1927681787051092526021693985, −22.47847877235416184190924618194, −20.832236268721342366203195058669, −19.683571029051085915028326002318, −18.93781650484556894554563674275, −17.92069036697696304678515954146, −17.19182920966900010784141220637, −15.94180803145919064173004334701, −14.307489569081785897762117761611, −13.03399776169197732960304998876, −11.772595554975361653931988204314, −10.89472568093878590305073934875, −9.868177906516226726156580268, −8.08404440694484874369696192779, −7.13244894493714496094751696036, −6.38614994152791165881008951030, −3.77050842561568468052912132880, −2.20848681130587870882685553616, −0.68103282877061852673423668231, 1.10299785430130081647531325560, 3.457482682283633718322389274603, 5.40772432391866305426394741390, 6.00291998043824359820924863015, 8.191290066120684236635084913203, 8.94887585233905966376813565268, 9.91691787644001295100330399909, 11.31571628605456538717756499282, 12.18753616120558302106683471953, 14.22110449601010978668640096272, 15.64054747734687664329949863702, 16.141708059945099791291679004563, 17.000109016060776869756137320, 18.28048010619362155034425152023, 19.39729984934440245879525825453, 20.66413003383889936282597818179, 21.36304059571013437333142674350, 22.880517703983729817507344444398, 24.01958590409244735223717363247, 25.09549671983219237541670179806, 25.954927454219915875489900226819, 27.426665508405233684334705522, 27.7233878216499640769934447228, 28.647661546938253832831156810, 29.55562651331683428814250895936

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