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Properties

Label 1-97-97.50-r0-0-0
Degree $1$
Conductor $97$
Sign $-0.160 + 0.986i$
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  − i·2-s + i·3-s − 4-s + (−0.707 + 0.707i)5-s + 6-s + (−0.707 − 0.707i)7-s + i·8-s − 9-s + (0.707 + 0.707i)10-s + i·11-s − i·12-s + (−0.707 + 0.707i)13-s + (−0.707 + 0.707i)14-s + (−0.707 − 0.707i)15-s + 16-s + (−0.707 + 0.707i)17-s + ⋯
L(s)  = 1  − i·2-s + i·3-s − 4-s + (−0.707 + 0.707i)5-s + 6-s + (−0.707 − 0.707i)7-s + i·8-s − 9-s + (0.707 + 0.707i)10-s + i·11-s − i·12-s + (−0.707 + 0.707i)13-s + (−0.707 + 0.707i)14-s + (−0.707 − 0.707i)15-s + 16-s + (−0.707 + 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.160 + 0.986i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ -0.160 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2849469840 + 0.3351898376i\)
\(L(\frac12)\) \(\approx\) \(0.2849469840 + 0.3351898376i\)
\(L(1)\) \(\approx\) \(0.6229177764 + 0.08559320446i\)
\(L(1)\) \(\approx\) \(0.6229177764 + 0.08559320446i\)

Euler product

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

−29.81745177960276199063054472462, −28.6951363695426295901819264307, −27.655129371230639256045989214652, −26.55531101419451172228744523928, −25.215263960304967975036905500558, −24.68047729336103941267470195226, −23.78140368716119303182310425452, −22.87079879491225651055735632549, −21.77330913385259464198825873636, −19.78237408197200763861116637633, −19.13727989787984502838833609725, −18.01263019084781118795648436711, −16.88359399225338362396977067613, −15.897620133251043467708817969246, −14.86219060414608797168650981030, −13.303014965075654396775126739503, −12.781293693384906176068355607539, −11.48087738263606445770388861047, −9.227596417163642426067469661096, −8.4170203204947688141204324012, −7.31721252827380365962570333068, −6.13077536705400197810044658785, −4.984121072741096695444393726362, −3.05662985153358830213493524301, −0.44388293747140658916412788350, 2.55717112398867477764972969279, 3.88251499778460117446938860994, 4.5759244586243782179070787767, 6.71467956216810900279428720378, 8.426686664871159536870790111473, 9.86819371207854202593236496218, 10.44105933254229017966108538387, 11.55942261517239798142272580181, 12.738741641231429029226473715311, 14.3003748651326923537153164493, 15.090602256661122726313583247663, 16.55561833798527699971515930800, 17.6248153929633298539243436452, 19.229678609468634077891452692020, 19.772351628580843078848287845157, 20.8645610345421409287719931804, 21.99917888399469564921615319593, 22.81135242599129625051053797454, 23.45878518486920357533835791269, 25.75563679270566062625081862685, 26.65132045511135853785536334814, 27.16136207788572907708633328075, 28.391075868669182484225024259766, 29.180479819436880439155992541325, 30.51509716334528827205591474357

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