Properties

Label 1-97-97.39-r1-0-0
Degree $1$
Conductor $97$
Sign $0.999 + 0.0150i$
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  + (−0.608 − 0.793i)2-s + (−0.130 + 0.991i)3-s + (−0.258 + 0.965i)4-s + (0.997 − 0.0654i)5-s + (0.866 − 0.5i)6-s + (−0.442 − 0.896i)7-s + (0.923 − 0.382i)8-s + (−0.965 − 0.258i)9-s + (−0.659 − 0.751i)10-s + (0.793 + 0.608i)11-s + (−0.923 − 0.382i)12-s + (−0.0654 − 0.997i)13-s + (−0.442 + 0.896i)14-s + (−0.0654 + 0.997i)15-s + (−0.866 − 0.5i)16-s + (0.896 + 0.442i)17-s + ⋯
L(s)  = 1  + (−0.608 − 0.793i)2-s + (−0.130 + 0.991i)3-s + (−0.258 + 0.965i)4-s + (0.997 − 0.0654i)5-s + (0.866 − 0.5i)6-s + (−0.442 − 0.896i)7-s + (0.923 − 0.382i)8-s + (−0.965 − 0.258i)9-s + (−0.659 − 0.751i)10-s + (0.793 + 0.608i)11-s + (−0.923 − 0.382i)12-s + (−0.0654 − 0.997i)13-s + (−0.442 + 0.896i)14-s + (−0.0654 + 0.997i)15-s + (−0.866 − 0.5i)16-s + (0.896 + 0.442i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.999 + 0.0150i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.999 + 0.0150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.367025943 + 0.01031625682i\)
\(L(\frac12)\) \(\approx\) \(1.367025943 + 0.01031625682i\)
\(L(1)\) \(\approx\) \(0.9401043563 - 0.05544355699i\)
\(L(1)\) \(\approx\) \(0.9401043563 - 0.05544355699i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.608 - 0.793i)T \)
3 \( 1 + (-0.130 + 0.991i)T \)
5 \( 1 + (0.997 - 0.0654i)T \)
7 \( 1 + (-0.442 - 0.896i)T \)
11 \( 1 + (0.793 + 0.608i)T \)
13 \( 1 + (-0.0654 - 0.997i)T \)
17 \( 1 + (0.896 + 0.442i)T \)
19 \( 1 + (0.555 + 0.831i)T \)
23 \( 1 + (0.321 + 0.946i)T \)
29 \( 1 + (0.659 - 0.751i)T \)
31 \( 1 + (-0.991 - 0.130i)T \)
37 \( 1 + (0.946 + 0.321i)T \)
41 \( 1 + (0.751 + 0.659i)T \)
43 \( 1 + (0.965 - 0.258i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.793 - 0.608i)T \)
59 \( 1 + (-0.321 + 0.946i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.831 + 0.555i)T \)
71 \( 1 + (0.751 - 0.659i)T \)
73 \( 1 + (-0.258 - 0.965i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (0.442 - 0.896i)T \)
89 \( 1 + (-0.923 + 0.382i)T \)
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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

−29.4309266137604486579553037618, −28.87331974042034949903735896162, −27.91194560194200387131713972037, −26.38944915504130670307994533106, −25.4419206863107047226441381750, −24.82007220260886146161675673047, −24.00645988145719015014207707872, −22.678622210271426672224052079941, −21.68540675298928094750103803081, −19.85309736833031279077854287049, −18.761478430447432082371997211333, −18.255213407898972972019918637859, −17.04656087994894219263067159812, −16.262068769998511967184199304372, −14.47523379190875584004870508860, −13.8891756559423753852770458585, −12.50358119089623290023637444512, −11.10709973227785377847786770513, −9.38808492966759312085383448286, −8.78569354391008121144369739216, −7.097791642403797486963445280330, −6.25609906219649347636270497930, −5.34709225880055543236987309492, −2.47032337440800139563494604562, −1.02351297498811146872250997136, 1.128926255086059167384210870218, 3.049971216856810686577580358045, 4.200741451082446338541225367124, 5.82592625221784580738942309453, 7.645386390221480708697579915172, 9.30154841722102877057233610752, 9.94721584311924291327370534007, 10.692432386643539688692731689534, 12.16811959214113856839795918363, 13.41944943662550118384872253671, 14.63133056095976174060718271160, 16.3491457154605454086138708451, 17.12056300501448664866221025842, 17.85835475761277978578666418345, 19.55963824385626350976821714452, 20.42220832425890443522212790105, 21.19930138368720191527782370948, 22.29441836192980524225997624310, 22.9930061855877351293709465766, 25.28367305613466237541687645185, 25.78273697292835058185863904203, 27.0122203071675924365712369957, 27.68892439825933459081183168314, 28.818490142453637827250994884228, 29.58556229698287379618573600939

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