Properties

Label 1-97-97.56-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

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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: $\chi_{97} (56, \cdot )$
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 \( 1 + (0.382 - 0.923i)T \)
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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

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

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