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Properties

Label 1-97-97.64-r0-0-0
Degree $1$
Conductor $97$
Sign $0.932 + 0.362i$
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  + i·2-s − i·3-s − 4-s + (0.707 + 0.707i)5-s + 6-s + (0.707 − 0.707i)7-s − i·8-s − 9-s + (−0.707 + 0.707i)10-s − i·11-s + i·12-s + (0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s + (0.707 − 0.707i)15-s + 16-s + (0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + i·2-s − i·3-s − 4-s + (0.707 + 0.707i)5-s + 6-s + (0.707 − 0.707i)7-s − i·8-s − 9-s + (−0.707 + 0.707i)10-s − i·11-s + i·12-s + (0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s + (0.707 − 0.707i)15-s + 16-s + (0.707 + 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.932 + 0.362i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ 0.932 + 0.362i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.044376660 + 0.1958752597i\)
\(L(\frac12)\) \(\approx\) \(1.044376660 + 0.1958752597i\)
\(L(1)\) \(\approx\) \(1.066507752 + 0.2014805902i\)
\(L(1)\) \(\approx\) \(1.066507752 + 0.2014805902i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 - iT \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 - iT \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 + iT \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 - T \)
79 \( 1 - iT \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 - iT \)
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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

−30.0060105101389191828405995545, −28.65459047155037410369622364487, −28.02488094909064972625614337620, −27.47304614679871393244362378353, −26.00693464497369342870837292550, −25.061096561368035529619642610843, −23.42687466677680465304282672397, −22.27286871040137716156385529169, −21.430126184690659207308172042126, −20.62149912817217465482428671356, −20.01459707208966278267368544444, −18.09832453748132351265632399947, −17.58811621661126708490002715672, −16.10009537195517818022900530260, −14.79362322278914975377062368311, −13.69693316462108742805501941352, −12.36444270070287330987346772026, −11.350064992202063891885912264700, −10.05164264962690883974882347278, −9.28992100781468445090754747768, −8.25998974725705302611186185642, −5.397856064461540323931627669608, −4.924123333895774484089549968335, −3.26884412299290330219973106736, −1.727638569288245472859623284640, 1.486164978769976875100306482428, 3.63194047332880704596419173217, 5.64983184417539074447895846963, 6.43298598096300756141079115508, 7.61895148028434825845026148420, 8.54533842877165449530887282002, 10.20632835452623836491294136044, 11.60268689209654380738508975290, 13.3560371860844900953501253817, 13.94383273222199018391716934990, 14.69867079916267751257584794087, 16.55755416084048682303167896587, 17.29535718368993796081357099227, 18.4686082642050132392792149563, 18.89165754576303574478892848782, 20.775108717541910323792897851594, 22.08116675515313728306203537139, 23.186684852659753601332549359749, 24.04956445375892754941613873846, 24.79992315443609733525002607928, 26.03481765030379629829761986962, 26.521557003604103892971448392190, 28.00166510351480691530955909927, 29.342832858871542331599968809905, 30.28920084318902386411444297043

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