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Properties

Label 1-97-97.84-r1-0-0
Degree $1$
Conductor $97$
Sign $-0.986 - 0.162i$
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 + (0.0654 − 0.997i)5-s + (0.866 + 0.5i)6-s + (−0.896 − 0.442i)7-s + (−0.923 − 0.382i)8-s + (−0.965 + 0.258i)9-s + (−0.751 − 0.659i)10-s + (−0.793 + 0.608i)11-s + (0.923 − 0.382i)12-s + (0.997 + 0.0654i)13-s + (−0.896 + 0.442i)14-s + (0.997 − 0.0654i)15-s + (−0.866 + 0.5i)16-s + (−0.442 − 0.896i)17-s + ⋯
L(s)  = 1  + (0.608 − 0.793i)2-s + (0.130 + 0.991i)3-s + (−0.258 − 0.965i)4-s + (0.0654 − 0.997i)5-s + (0.866 + 0.5i)6-s + (−0.896 − 0.442i)7-s + (−0.923 − 0.382i)8-s + (−0.965 + 0.258i)9-s + (−0.751 − 0.659i)10-s + (−0.793 + 0.608i)11-s + (0.923 − 0.382i)12-s + (0.997 + 0.0654i)13-s + (−0.896 + 0.442i)14-s + (0.997 − 0.0654i)15-s + (−0.866 + 0.5i)16-s + (−0.442 − 0.896i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.986 - 0.162i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ -0.986 - 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08788106392 - 1.072795636i\)
\(L(\frac12)\) \(\approx\) \(0.08788106392 - 1.072795636i\)
\(L(1)\) \(\approx\) \(0.8983502998 - 0.5521758387i\)
\(L(1)\) \(\approx\) \(0.8983502998 - 0.5521758387i\)

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 \( 1 + (0.0654 - 0.997i)T \)
7 \( 1 + (-0.896 - 0.442i)T \)
11 \( 1 + (-0.793 + 0.608i)T \)
13 \( 1 + (0.997 + 0.0654i)T \)
17 \( 1 + (-0.442 - 0.896i)T \)
19 \( 1 + (-0.831 - 0.555i)T \)
23 \( 1 + (-0.946 - 0.321i)T \)
29 \( 1 + (0.751 - 0.659i)T \)
31 \( 1 + (0.991 - 0.130i)T \)
37 \( 1 + (0.321 + 0.946i)T \)
41 \( 1 + (-0.659 - 0.751i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.793 - 0.608i)T \)
59 \( 1 + (0.946 - 0.321i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.555 + 0.831i)T \)
71 \( 1 + (-0.659 + 0.751i)T \)
73 \( 1 + (-0.258 + 0.965i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (0.896 - 0.442i)T \)
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

−30.416418714063817461296172825844, −29.67650910751365292636415409016, −28.523330728909820784813292752, −26.666639539041811033795633872915, −25.7677228719349824475431558480, −25.33668436701449358686869856661, −23.88409457990772549538685582863, −23.2327073328152954398335925742, −22.254612743269546770341258694957, −21.1979966502972764481655250523, −19.42502030144788876877514245343, −18.498367193105551431379548396989, −17.6832471701826050672692258001, −16.18083660908077678372580592348, −15.18483656869270121935801066086, −13.97084753172383825465989039547, −13.193614687585188562129946783233, −12.17685359827671190009932311605, −10.71382436175916548509752044803, −8.71845373010983654066452379888, −7.727348561560650802583160016474, −6.263632458083938723632581153186, −6.06098093931283590744458305681, −3.59646872739423276068856765520, −2.52874992670300710733310783892, 0.36277053016058231720481824588, 2.542858591633289800941445126510, 4.02262504735672398908641485852, 4.83487408832817830423404917215, 6.193863403379895663429843680877, 8.56051225391405629198602985684, 9.672681204462462365136346990579, 10.46347676826918203214310460548, 11.81111635228907927478033598337, 13.10599653286931721128604749477, 13.81693825665176763688628614097, 15.52859300226234114069885477588, 16.03536708177161433837811454634, 17.53869452775498059401595002361, 19.1768770171071889852290659785, 20.38948795265066843707762305907, 20.6467215486789388908445937997, 21.85103844041568425521368519020, 22.929710381529573124228982010249, 23.68761986478560880089334160169, 25.25624072398300513775171239328, 26.33878643331269606546041208200, 27.65394545162715836259128862925, 28.46597346033239718024693998324, 29.04227691752527239774326067611

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