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Properties

Label 1-97-97.3-r0-0-0
Degree $1$
Conductor $97$
Sign $-0.285 - 0.958i$
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  + (0.258 − 0.965i)2-s + (0.965 + 0.258i)3-s + (−0.866 − 0.5i)4-s + (−0.130 − 0.991i)5-s + (0.5 − 0.866i)6-s + (−0.793 − 0.608i)7-s + (−0.707 + 0.707i)8-s + (0.866 + 0.5i)9-s + (−0.991 − 0.130i)10-s + (−0.258 − 0.965i)11-s + (−0.707 − 0.707i)12-s + (0.130 + 0.991i)13-s + (−0.793 + 0.608i)14-s + (0.130 − 0.991i)15-s + (0.5 + 0.866i)16-s + (0.793 − 0.608i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (0.965 + 0.258i)3-s + (−0.866 − 0.5i)4-s + (−0.130 − 0.991i)5-s + (0.5 − 0.866i)6-s + (−0.793 − 0.608i)7-s + (−0.707 + 0.707i)8-s + (0.866 + 0.5i)9-s + (−0.991 − 0.130i)10-s + (−0.258 − 0.965i)11-s + (−0.707 − 0.707i)12-s + (0.130 + 0.991i)13-s + (−0.793 + 0.608i)14-s + (0.130 − 0.991i)15-s + (0.5 + 0.866i)16-s + (0.793 − 0.608i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.285 - 0.958i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ -0.285 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7640792677 - 1.024992037i\)
\(L(\frac12)\) \(\approx\) \(0.7640792677 - 1.024992037i\)
\(L(1)\) \(\approx\) \(1.039618606 - 0.7725182109i\)
\(L(1)\) \(\approx\) \(1.039618606 - 0.7725182109i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (-0.130 - 0.991i)T \)
7 \( 1 + (-0.793 - 0.608i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (0.130 + 0.991i)T \)
17 \( 1 + (0.793 - 0.608i)T \)
19 \( 1 + (0.923 + 0.382i)T \)
23 \( 1 + (0.608 + 0.793i)T \)
29 \( 1 + (-0.991 + 0.130i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (-0.608 + 0.793i)T \)
41 \( 1 + (0.991 - 0.130i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (0.608 - 0.793i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.923 - 0.382i)T \)
71 \( 1 + (0.991 + 0.130i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.793 + 0.608i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
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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

−30.73038552691628370395222674105, −29.8202582403662282777026894397, −28.06351454635200320280335059470, −26.81832317403899083623526537071, −25.86993866363651389861260859704, −25.483890878040406564861334750403, −24.396197021075948413671316225, −23.006172364660217648232067866387, −22.38159868944875534230473702311, −21.07516417981639340300930789746, −19.609635472934713183992728437574, −18.540889779186952485905862179899, −17.85595582876948010491618424188, −16.05591062883007971592688331997, −15.09835815158883911549679486756, −14.58497271936000284664297682395, −13.177267954436138430076195453, −12.41734235480736865940902881540, −10.17288996120703979467880271501, −9.12637465002292486247822538417, −7.73372604244815106395272257052, −6.994574609515198506620142376705, −5.646642903587259774096804294361, −3.709206970227403374128488303911, −2.727992158056818687966467706036, 1.32500438393666703535598228375, 3.15717615608873796803270922799, 4.030383358005564817824216063326, 5.44588680398148116340782991863, 7.66182637752579129238961919509, 9.113477682026989416284725745270, 9.60825808694155467961815590227, 11.10351933742670669422631324262, 12.50537788261977186375374065018, 13.51956471336542865861850189665, 14.12217835324527016342168870729, 15.84888069289387623586124354773, 16.75580883694303073367039310586, 18.73461550637458354996663036062, 19.36590108556361021366230769257, 20.45613013156403940716608015903, 20.993849556969688778090474114140, 22.13695463177122531403087037402, 23.52814945922974916855701260186, 24.40057651568523960749883940105, 25.84480899699349776331612151620, 26.86817935362448213485705368430, 27.69570129090059820641663192291, 29.05454512171974293329525407447, 29.60167129159095290987281267997

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