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Properties

Label 1-97-97.37-r1-0-0
Degree $1$
Conductor $97$
Sign $0.527 - 0.849i$
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  + (0.130 + 0.991i)2-s + (−0.608 + 0.793i)3-s + (−0.965 + 0.258i)4-s + (0.946 − 0.321i)5-s + (−0.866 − 0.5i)6-s + (−0.751 + 0.659i)7-s + (−0.382 − 0.923i)8-s + (−0.258 − 0.965i)9-s + (0.442 + 0.896i)10-s + (−0.991 − 0.130i)11-s + (0.382 − 0.923i)12-s + (−0.321 − 0.946i)13-s + (−0.751 − 0.659i)14-s + (−0.321 + 0.946i)15-s + (0.866 − 0.5i)16-s + (−0.659 + 0.751i)17-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.608 + 0.793i)3-s + (−0.965 + 0.258i)4-s + (0.946 − 0.321i)5-s + (−0.866 − 0.5i)6-s + (−0.751 + 0.659i)7-s + (−0.382 − 0.923i)8-s + (−0.258 − 0.965i)9-s + (0.442 + 0.896i)10-s + (−0.991 − 0.130i)11-s + (0.382 − 0.923i)12-s + (−0.321 − 0.946i)13-s + (−0.751 − 0.659i)14-s + (−0.321 + 0.946i)15-s + (0.866 − 0.5i)16-s + (−0.659 + 0.751i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.527 - 0.849i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.527 - 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1782400627 - 0.09916343835i\)
\(L(\frac12)\) \(\approx\) \(0.1782400627 - 0.09916343835i\)
\(L(1)\) \(\approx\) \(0.5467738276 + 0.3786765569i\)
\(L(1)\) \(\approx\) \(0.5467738276 + 0.3786765569i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (0.130 + 0.991i)T \)
3 \( 1 + (-0.608 + 0.793i)T \)
5 \( 1 + (0.946 - 0.321i)T \)
7 \( 1 + (-0.751 + 0.659i)T \)
11 \( 1 + (-0.991 - 0.130i)T \)
13 \( 1 + (-0.321 - 0.946i)T \)
17 \( 1 + (-0.659 + 0.751i)T \)
19 \( 1 + (0.195 - 0.980i)T \)
23 \( 1 + (0.997 - 0.0654i)T \)
29 \( 1 + (-0.442 + 0.896i)T \)
31 \( 1 + (-0.793 - 0.608i)T \)
37 \( 1 + (-0.0654 + 0.997i)T \)
41 \( 1 + (-0.896 - 0.442i)T \)
43 \( 1 + (0.258 - 0.965i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.991 + 0.130i)T \)
59 \( 1 + (-0.997 - 0.0654i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.980 + 0.195i)T \)
71 \( 1 + (-0.896 + 0.442i)T \)
73 \( 1 + (-0.965 - 0.258i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (0.751 + 0.659i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
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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

−29.65895764017031158832034662220, −29.027378458963917499164347610626, −28.6810726564805941101192281396, −26.96294237472363604621769745046, −26.0086185366420828402621748811, −24.65109841306361781478061522156, −23.32086425354274595319097436899, −22.71426899409807643128447654340, −21.637623513876788230171105437833, −20.55248987260121844684121358966, −19.233891934536554700552221068321, −18.46017844269968471708575903186, −17.53178407492016772544970174919, −16.452233938455248746587836316788, −14.27867377421786982709912867336, −13.3767521947560352265854759372, −12.72301790886637773818183887692, −11.30779693623353913847484097547, −10.356889211623855382279736513893, −9.31668968233954109669325450573, −7.387410422595222814910274718920, −6.10088180114154922468538465901, −4.83047647826814182865274840324, −2.87315076903493495310550960665, −1.63750417597053792489923284673, 0.091704341440684167326168707312, 3.10434015370990138802925403063, 4.99447843777028789783444194732, 5.599253262473253656912684271127, 6.72921047064234541765015251502, 8.64992128481925925558882808035, 9.543007735043958674471855062769, 10.61535336865488060476010487939, 12.62415221513720107157414761434, 13.28618635086138056698449147059, 15.01980173669223325033141849905, 15.642981336730355779495272553121, 16.76821063318101276764668796190, 17.58463730404576891022699607494, 18.55436607235139281531381902643, 20.46431869370541544240680168664, 21.91227414339546765932468824367, 22.05135606651411310973976523353, 23.41930254447723256042056148780, 24.487039836863356744926423000459, 25.682613250897479967683073914460, 26.24910600625016928244136509516, 27.566296267769222548859703216324, 28.526229950579280700103595582596, 29.3155822267755771700318710632

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