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Properties

Label 1-97-97.27-r0-0-0
Degree $1$
Conductor $97$
Sign $-0.0486 + 0.998i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0486 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0486 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.0486 + 0.998i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ -0.0486 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6635714228 + 0.6966525242i\)
\(L(\frac12)\) \(\approx\) \(0.6635714228 + 0.6966525242i\)
\(L(1)\) \(\approx\) \(0.8292624504 + 0.5304617907i\)
\(L(1)\) \(\approx\) \(0.8292624504 + 0.5304617907i\)

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

−29.75925567333492757626837437708, −28.72390754650391978190683656067, −28.04765390382270031208137642705, −26.68950245835052688997104881584, −25.74307258162342457987635919812, −24.62154268147794263929670140281, −24.15382862669839461280514097885, −21.88320387783206683438138580623, −21.298072929050834764110718927936, −20.03709292814191507164331983073, −19.36728168936878768297765223890, −18.261729411894593787316052703154, −17.34990837338381108486282229922, −16.14905129870592348474881439351, −14.46089834892140406284147157942, −13.23282751597224192316498910238, −12.30260720933460208643142613268, −11.38980450761918144478955917664, −9.32944252163572748684359047129, −8.933910780714109640745696637378, −7.86178705713009017605959455933, −6.21272097674742442260148895559, −4.181102642154717347201579180919, −2.44425973129294702577620413051, −1.42424761721088411061182516599, 2.05025067539745439191742840425, 3.889465711054624131419815135916, 5.43790404037973794095028211766, 7.12004368063868706535417249067, 7.8552940614757744428893131345, 9.50922390743849164704351272572, 10.11899425135237404366041725345, 11.17291795379107930928733238, 13.64780732483729480562617177327, 14.478412085917524888864397045924, 15.15964467399975329404166496834, 16.47517316450189496952207731392, 17.52333832230022617729391634820, 18.509666923888102186386022538425, 19.93429397633467418410409920976, 20.434811608950944189587028804936, 22.202987429283951341343987228655, 22.94143832395573040500156432147, 24.542793782147726802592123386604, 25.43871712012455923740443959595, 26.22082981262359043762837255012, 27.17736489934507641243826333153, 27.637818687680838942685756234820, 29.36036750644000330700385790599, 30.28912443694643183419912464213

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