↑

Properties

Label 1-97-97.26-r1-0-0
Degree $1$
Conductor $97$
Sign $0.0711 - 0.997i$
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  + (0.793 − 0.608i)2-s + (0.991 + 0.130i)3-s + (0.258 − 0.965i)4-s + (−0.751 − 0.659i)5-s + (0.866 − 0.5i)6-s + (0.946 + 0.321i)7-s + (−0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (−0.997 − 0.0654i)10-s + (0.608 − 0.793i)11-s + (0.382 − 0.923i)12-s + (−0.659 + 0.751i)13-s + (0.946 − 0.321i)14-s + (−0.659 − 0.751i)15-s + (−0.866 − 0.5i)16-s + (−0.321 − 0.946i)17-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + (0.991 + 0.130i)3-s + (0.258 − 0.965i)4-s + (−0.751 − 0.659i)5-s + (0.866 − 0.5i)6-s + (0.946 + 0.321i)7-s + (−0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (−0.997 − 0.0654i)10-s + (0.608 − 0.793i)11-s + (0.382 − 0.923i)12-s + (−0.659 + 0.751i)13-s + (0.946 − 0.321i)14-s + (−0.659 − 0.751i)15-s + (−0.866 − 0.5i)16-s + (−0.321 − 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0711 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0711 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.0711 - 0.997i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.0711 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.566377588 - 2.389804748i\)
\(L(\frac12)\) \(\approx\) \(2.566377588 - 2.389804748i\)
\(L(1)\) \(\approx\) \(1.910776600 - 1.032425283i\)
\(L(1)\) \(\approx\) \(1.910776600 - 1.032425283i\)

Euler product

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

−30.65488685718328289924048366148, −29.70872753674888973451405654442, −27.48500615904997898110398250727, −26.801457250140287219844255069368, −25.80076723638343334010511896091, −24.7624122117959775042844061484, −23.97008855248296281879004455796, −22.858452404326433515161042399274, −21.82695555525794537421086864821, −20.51989105788541249300822887058, −19.85157376892391103099553703394, −18.33097273972279175613960688146, −17.18386153294650387009592571964, −15.61249914245203671147752384301, −14.66703479643336263704629721463, −14.39302006513342960675610756268, −12.81791572621635047876260892647, −11.8209561542628999583622152159, −10.262296889249747401376221738, −8.287907921680189117081697609088, −7.711393575848332159575496806768, −6.58930388684537665845805908099, −4.57070528304514631358299543447, −3.6813945527209434745507590745, −2.20201023800986516795992536586, 1.28241226385783953275788096861, 2.792872056934194360186715710630, 4.190418725850727422444768882016, 5.028313485623048409641205773709, 7.107664783971295561400270444557, 8.568068032772781256454196164946, 9.50648099402571333890149571917, 11.30379255980877432432419644139, 11.99072212744217144008061397045, 13.44451496819156849577055704101, 14.28713298404360560757866576983, 15.28555674096919913723741550776, 16.28071132827626755176530509934, 18.273930226758349578883473281362, 19.64820653131238338799007536066, 19.89784532280111705658308486347, 21.297645472117740126977647979508, 21.75431056525706574590315659411, 23.49279988221505508200225647354, 24.40304189187732333762072709001, 24.89516885607498277613864617290, 26.86315484247231050995654570822, 27.444539004459006899652749251, 28.61860864015568347989951951716, 29.94478091343079207256961130534

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