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Properties

Label 1-97-97.40-r1-0-0
Degree $1$
Conductor $97$
Sign $-0.312 + 0.949i$
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  + (−0.991 + 0.130i)2-s + (0.793 + 0.608i)3-s + (0.965 − 0.258i)4-s + (0.896 + 0.442i)5-s + (−0.866 − 0.5i)6-s + (−0.0654 + 0.997i)7-s + (−0.923 + 0.382i)8-s + (0.258 + 0.965i)9-s + (−0.946 − 0.321i)10-s + (−0.130 + 0.991i)11-s + (0.923 + 0.382i)12-s + (0.442 − 0.896i)13-s + (−0.0654 − 0.997i)14-s + (0.442 + 0.896i)15-s + (0.866 − 0.5i)16-s + (−0.997 + 0.0654i)17-s + ⋯
L(s)  = 1  + (−0.991 + 0.130i)2-s + (0.793 + 0.608i)3-s + (0.965 − 0.258i)4-s + (0.896 + 0.442i)5-s + (−0.866 − 0.5i)6-s + (−0.0654 + 0.997i)7-s + (−0.923 + 0.382i)8-s + (0.258 + 0.965i)9-s + (−0.946 − 0.321i)10-s + (−0.130 + 0.991i)11-s + (0.923 + 0.382i)12-s + (0.442 − 0.896i)13-s + (−0.0654 − 0.997i)14-s + (0.442 + 0.896i)15-s + (0.866 − 0.5i)16-s + (−0.997 + 0.0654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.312 + 0.949i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ -0.312 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9293770458 + 1.284072908i\)
\(L(\frac12)\) \(\approx\) \(0.9293770458 + 1.284072908i\)
\(L(1)\) \(\approx\) \(0.9290354986 + 0.5320882069i\)
\(L(1)\) \(\approx\) \(0.9290354986 + 0.5320882069i\)

Euler product

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

−29.28621151290284966852641983403, −28.89162038308395250096442562636, −27.18349771420640982276423441143, −26.29178037756109820768115437584, −25.63861470206036145508706353340, −24.42479397155750715678369557231, −23.8975896027431228992188996382, −21.71442006785577744054885943021, −20.66757377940709171795708008857, −19.99936701810956358796870655414, −18.8549051856058725811667629938, −17.92841274866457608555948713886, −16.88849439958592678037271307513, −15.85701640190595942399574137098, −14.00467851376937291645750138009, −13.44623095774921770336698902043, −11.91830907757062622637676786662, −10.48349084303892247885500620753, −9.31868680091801231496199459033, −8.47674485017837761293774397056, −7.20343880186537649495774158795, −6.11859024786406633363707842548, −3.64373162049958775786820392050, −2.045163861027081637764963434062, −0.89899368491977522025938143715, 2.0110079306723282369830615838, 2.898659544267440755836282439522, 5.22132093572700323733679608974, 6.62466768278861398642814500232, 8.10445630132372670172860660459, 9.186552027114005441790384266173, 9.9546459832633597997038925167, 10.99012096238930163892178231555, 12.73523578550753749920919853045, 14.28733683611854945303160339335, 15.31397563500441152683262387312, 15.99180858678830658524625508047, 17.74125208999866456432212683038, 18.19674434690603622577345121301, 19.611123263875774348925926360279, 20.494205303710455982853376628157, 21.50788543038763151819758921441, 22.50778648769822557284975694709, 24.60471739017703318468498199306, 25.24919356906548940262403486873, 25.976778200880488964180874813626, 26.84972713481668962965770106937, 28.12781523512568182818587516711, 28.668349459222484281270933907017, 30.19118072269472008721832863042

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