↑

Properties

Label 1-97-97.74-r1-0-0
Degree $1$
Conductor $97$
Sign $-0.535 - 0.844i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.130 + 0.991i)2-s + (0.608 + 0.793i)3-s + (−0.965 − 0.258i)4-s + (−0.321 + 0.946i)5-s + (−0.866 + 0.5i)6-s + (−0.659 + 0.751i)7-s + (0.382 − 0.923i)8-s + (−0.258 + 0.965i)9-s + (−0.896 − 0.442i)10-s + (0.991 − 0.130i)11-s + (−0.382 − 0.923i)12-s + (−0.946 − 0.321i)13-s + (−0.659 − 0.751i)14-s + (−0.946 + 0.321i)15-s + (0.866 + 0.5i)16-s + (0.751 − 0.659i)17-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (0.608 + 0.793i)3-s + (−0.965 − 0.258i)4-s + (−0.321 + 0.946i)5-s + (−0.866 + 0.5i)6-s + (−0.659 + 0.751i)7-s + (0.382 − 0.923i)8-s + (−0.258 + 0.965i)9-s + (−0.896 − 0.442i)10-s + (0.991 − 0.130i)11-s + (−0.382 − 0.923i)12-s + (−0.946 − 0.321i)13-s + (−0.659 − 0.751i)14-s + (−0.946 + 0.321i)15-s + (0.866 + 0.5i)16-s + (0.751 − 0.659i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.535 - 0.844i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ -0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5007805212 + 0.9103191966i\)
\(L(\frac12)\) \(\approx\) \(-0.5007805212 + 0.9103191966i\)
\(L(1)\) \(\approx\) \(0.4327643782 + 0.7891595847i\)
\(L(1)\) \(\approx\) \(0.4327643782 + 0.7891595847i\)

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

Imaginary part of the first few zeros on the critical line

−29.337550469962682389316073404982, −28.333970834475488489815561007426, −27.19147812071252041668322798782, −26.18139108335894466103626832394, −24.99624229576741929670478297878, −23.79523766067362379550273954216, −22.96128280634151946303885828612, −21.49018893631461038224751678893, −20.3328067269715166758720555062, −19.58258012949740616573866711276, −19.11939999559753732778542874981, −17.41600564499365622613966872587, −16.73576139077717674548406202272, −14.65789507438100169026456664424, −13.63064089136509982303859908875, −12.50252768417405378777752060397, −12.078535492222867367119993609098, −10.228297758041830585256935549, −9.08477248377812194824059270599, −8.16236779793808604188456029630, −6.70447514552091122223900939071, −4.53546871803094386595414827693, −3.40496339898200625006313411477, −1.72509960138381734282043492139, −0.45900167332815919977253205206, 2.86422939209670167698324337761, 4.11105996142828458359615897086, 5.66642128308457712046842771688, 6.95859341317668253054966963881, 8.20281459958743733499238314837, 9.43159802170147471758568829431, 10.19873080575857526141633311200, 11.98349403490263694162302979332, 13.7272088676317509545729015810, 14.71804801978283526456492015159, 15.324221224667356279686251404029, 16.33894094148402779812518764103, 17.5373378084162218457722269227, 19.15080875788367522710877017520, 19.38528411142801853122213650730, 21.44477448996797089957059023044, 22.33177429042708068567018566569, 22.95429413547714761538252199371, 24.68926321917740835153219485722, 25.46629868345025844091705685534, 26.216426421790392779482087534030, 27.37087104948002432184232098112, 27.71923165111849132903294939699, 29.575590133281101985097274137021, 30.93027059972099320272810108155

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