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Properties

Label 1-97-97.48-r0-0-0
Degree $1$
Conductor $97$
Sign $0.860 + 0.508i$
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  + (0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 − 0.5i)4-s + (0.608 + 0.793i)5-s + (0.5 + 0.866i)6-s + (−0.991 − 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.793 + 0.608i)10-s + (−0.965 − 0.258i)11-s + (0.707 + 0.707i)12-s + (−0.608 − 0.793i)13-s + (−0.991 + 0.130i)14-s + (−0.608 + 0.793i)15-s + (0.5 − 0.866i)16-s + (0.991 − 0.130i)17-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 − 0.5i)4-s + (0.608 + 0.793i)5-s + (0.5 + 0.866i)6-s + (−0.991 − 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.793 + 0.608i)10-s + (−0.965 − 0.258i)11-s + (0.707 + 0.707i)12-s + (−0.608 − 0.793i)13-s + (−0.991 + 0.130i)14-s + (−0.608 + 0.793i)15-s + (0.5 − 0.866i)16-s + (0.991 − 0.130i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.860 + 0.508i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ 0.860 + 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.718914927 + 0.4698791485i\)
\(L(\frac12)\) \(\approx\) \(1.718914927 + 0.4698791485i\)
\(L(1)\) \(\approx\) \(1.721483205 + 0.2979663345i\)
\(L(1)\) \(\approx\) \(1.721483205 + 0.2979663345i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.608 + 0.793i)T \)
7 \( 1 + (-0.991 - 0.130i)T \)
11 \( 1 + (-0.965 - 0.258i)T \)
13 \( 1 + (-0.608 - 0.793i)T \)
17 \( 1 + (0.991 - 0.130i)T \)
19 \( 1 + (0.382 - 0.923i)T \)
23 \( 1 + (0.130 + 0.991i)T \)
29 \( 1 + (0.793 - 0.608i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (-0.130 + 0.991i)T \)
41 \( 1 + (-0.793 + 0.608i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (0.130 - 0.991i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.382 + 0.923i)T \)
71 \( 1 + (-0.793 - 0.608i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (-0.991 + 0.130i)T \)
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

−30.03405465929483439965684813943, −28.91142815378259587529120256915, −28.81345396440999593725429982349, −26.346293245933125943140762560505, −25.429516366410467005989594117585, −24.808004143388127966644176538813, −23.72408603917189047058564501841, −22.98313068105265669746481783605, −21.60594664584425009108297557729, −20.622385832847689033381207060724, −19.61978951862054658071571072969, −18.35171651479661812373943623546, −16.88849380085109803367527478644, −16.14041791120127677499501349651, −14.51578649878212733947262102252, −13.64211627611770134496016505122, −12.52984818136160952043458517719, −12.26176186562112670965789359240, −10.1388986007963550367917147334, −8.57370387947060627174618124737, −7.26721893246254251997102134289, −6.15363932376450579208177009024, −5.07849153287870071247515871787, −3.18536992412232966565117024243, −1.89999181132362608181966129234, 2.7635634161126596312508867674, 3.267574842240073204288825746048, 5.05187189119031285576852841266, 6.009718988875711604476225360068, 7.53140386272137829602768694341, 9.79290349419283269696941203459, 10.2168490883041642058357458295, 11.45312095747561503400474483700, 13.10880867340335101108887055888, 13.88946311788484686549824356583, 15.12915023486368391376323917241, 15.772741736791380048153914595122, 17.084530575123921464286724575846, 18.84946479188500875915288850987, 19.9033087163886234975741334330, 20.98034482744242178529139534905, 21.909564934651511601103340024439, 22.52047552073711727914360427481, 23.53374614986010358307589351761, 25.27058874086582269780213689468, 25.81533073108637366715382574542, 26.9469937660029743460492464900, 28.41715571963721873819554387325, 29.29051592224269356974916397763, 30.17861796004479086843573706832

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