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Properties

Label 1-97-97.70-r0-0-0
Degree $1$
Conductor $97$
Sign $-0.112 + 0.993i$
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  + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − i·4-s + (−0.382 − 0.923i)5-s − 6-s + (−0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + i·9-s + (0.923 + 0.382i)10-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s + (0.382 + 0.923i)13-s + (−0.382 − 0.923i)14-s + (0.382 − 0.923i)15-s − 16-s + (0.382 + 0.923i)17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − i·4-s + (−0.382 − 0.923i)5-s − 6-s + (−0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + i·9-s + (0.923 + 0.382i)10-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s + (0.382 + 0.923i)13-s + (−0.382 − 0.923i)14-s + (0.382 − 0.923i)15-s − 16-s + (0.382 + 0.923i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.112 + 0.993i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ -0.112 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5468672172 + 0.6124708509i\)
\(L(\frac12)\) \(\approx\) \(0.5468672172 + 0.6124708509i\)
\(L(1)\) \(\approx\) \(0.7389995239 + 0.4514630738i\)
\(L(1)\) \(\approx\) \(0.7389995239 + 0.4514630738i\)

Euler product

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

−29.624695197084017894178411622956, −29.39667179914159584429856774604, −27.28809108008035273326987124818, −26.97652353281210637404118206344, −25.7319473986132097211395255230, −25.09169178433631472680192126065, −23.37519224836013014326003599473, −22.511790680159391254709447734538, −21.01074261707408316441092709462, −19.981829346212419897247106693786, −19.24535208131778315362174473848, −18.44931110759888941210350902346, −17.36720305281047714266309272878, −16.00704805742049875398491183936, −14.3955044817768849774751214174, −13.464308576349751185568601801834, −12.237496679174009640151254333738, −11.02600169889289495889321079860, −9.976494260653600296376293492399, −8.58020245632630691121972181618, −7.50900325100954554015083243049, −6.63603505904807436097936057252, −3.61994440142703590956172682903, −3.04170251869705487163445011786, −1.11957195672536038071649626917, 1.94227475583114302831038365624, 4.116106440910398916183212906288, 5.30768987175230816170642716820, 6.90973259454634705975935195360, 8.596878160854882804773013870906, 8.90085960503842699538344251132, 10.02482792055360428397519922452, 11.63317863703960888316269426564, 13.18883479574082047026845296335, 14.69739823142842460285531624069, 15.41914490857494887064855717952, 16.36371602574265864105586690497, 17.24839379453181227644584013783, 18.97897659434291207121241743715, 19.565054959622627805127952884, 20.689224093550347184479657109540, 21.87904286730920386072160154030, 23.31294725252268036129649043173, 24.50000794639883848646981412749, 25.34390875323556899276849999789, 26.076506882598696663545100675103, 27.309704432852860317972191187106, 28.13363311360398328431372232358, 28.63504213677375978399141972526, 30.734408999232075980790504921667

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