↑

Properties

Label 1-97-97.88-r0-0-0
Degree $1$
Conductor $97$
Sign $0.963 - 0.267i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.965 − 0.258i)5-s + (−0.5 − 0.866i)6-s + (−0.258 − 0.965i)7-s − i·8-s + (0.5 + 0.866i)9-s + (−0.965 − 0.258i)10-s + (−0.866 + 0.5i)11-s + i·12-s + (0.965 − 0.258i)13-s + (−0.258 + 0.965i)14-s + (0.965 + 0.258i)15-s + (−0.5 + 0.866i)16-s + (−0.258 + 0.965i)17-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.965 − 0.258i)5-s + (−0.5 − 0.866i)6-s + (−0.258 − 0.965i)7-s − i·8-s + (0.5 + 0.866i)9-s + (−0.965 − 0.258i)10-s + (−0.866 + 0.5i)11-s + i·12-s + (0.965 − 0.258i)13-s + (−0.258 + 0.965i)14-s + (0.965 + 0.258i)15-s + (−0.5 + 0.866i)16-s + (−0.258 + 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.963 - 0.267i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ 0.963 - 0.267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9853111048 - 0.1343368917i\)
\(L(\frac12)\) \(\approx\) \(0.9853111048 - 0.1343368917i\)
\(L(1)\) \(\approx\) \(1.003404380 - 0.1093913019i\)
\(L(1)\) \(\approx\) \(1.003404380 - 0.1093913019i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.965 - 0.258i)T \)
17 \( 1 + (-0.258 + 0.965i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.258 - 0.965i)T \)
29 \( 1 + (-0.965 + 0.258i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 + (-0.965 + 0.258i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.258 + 0.965i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + (-0.965 - 0.258i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.258 + 0.965i)T \)
89 \( 1 - iT \)
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

−29.801382917885031535657641136342, −29.13115641852226423457094955086, −28.08369948989473607768132919101, −26.653376299947225055883993085496, −25.828365910255135659476992963624, −25.19617482971941260652929208646, −24.40732864212808141403141710828, −23.12404449163847864035849348337, −21.3412871805248938694640113610, −20.61388410383122801010270278844, −18.99492831216557407287287947099, −18.586563450741299471078888321391, −17.681391616611533245526741374097, −16.117926287198228977542472223, −15.16451946978957870903563371398, −14.01117585875615956176906194122, −13.052531801012146632092495557697, −11.26137041838283357935076294137, −9.77505474104878204142000729785, −8.98619379544047487153011462130, −7.948276750149101086941501785233, −6.52503005108132817918184120152, −5.624928872084741442131660603446, −2.86188965815523376177020601848, −1.76809608108983182253993744389, 1.69076615044591364026996799024, 3.02775987077849776336868586648, 4.473238877705848048372711945068, 6.63593325950480603406216145998, 8.091549569283145349388396659267, 9.02063171757745193906872979286, 10.292907971252227915318890933011, 10.63206566651824580886862782213, 12.961887994840210246885315599430, 13.423164100125594519062121738330, 15.11493922184475488597632102588, 16.34241332447371242301403577733, 17.283301401215321518462586673378, 18.42788373069709696807605911764, 19.64606104927285565016034638981, 20.640622007645208145398371334970, 21.05868148108642396792471748121, 22.3098413600125520086469822705, 24.06489851326053176537368691959, 25.559644663221464351469067438986, 25.87993063896290584263001612722, 26.758789552635869960951367987511, 28.07243136216865935715090960743, 28.80706244564876446669288080175, 30.128467302487365376462224927995

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