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Properties

Label 1-97-97.49-r0-0-0
Degree $1$
Conductor $97$
Sign $0.998 + 0.0517i$
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  + (0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 − 0.5i)4-s + (−0.608 − 0.793i)5-s + (0.5 + 0.866i)6-s + (0.991 + 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (−0.793 − 0.608i)10-s + (−0.965 − 0.258i)11-s + (0.707 + 0.707i)12-s + (0.608 + 0.793i)13-s + (0.991 − 0.130i)14-s + (0.608 − 0.793i)15-s + (0.5 − 0.866i)16-s + (−0.991 + 0.130i)17-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 − 0.5i)4-s + (−0.608 − 0.793i)5-s + (0.5 + 0.866i)6-s + (0.991 + 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (−0.793 − 0.608i)10-s + (−0.965 − 0.258i)11-s + (0.707 + 0.707i)12-s + (0.608 + 0.793i)13-s + (0.991 − 0.130i)14-s + (0.608 − 0.793i)15-s + (0.5 − 0.866i)16-s + (−0.991 + 0.130i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.998 + 0.0517i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ 0.998 + 0.0517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.747529459 + 0.04528578014i\)
\(L(\frac12)\) \(\approx\) \(1.747529459 + 0.04528578014i\)
\(L(1)\) \(\approx\) \(1.715567167 + 0.02660170940i\)
\(L(1)\) \(\approx\) \(1.715567167 + 0.02660170940i\)

Euler product

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

−30.3721687580719247025844057860, −29.60739375722326355444804952544, −28.21848167221116012654042440181, −26.59841285296253857434425469865, −25.73910054264005996932855718249, −24.63558483658326344606940319009, −23.61383243853434414837780547310, −23.20136955505719293596394397208, −21.861996754832148512490057972999, −20.55120171777560888300209926396, −19.73114496794814755700373138376, −18.26059396461625821640917734105, −17.50634764060209825252828298811, −15.57948263802039994958803038056, −14.953095727393452662089121971353, −13.739364770798918585270609321546, −12.945688418124425393808933025552, −11.53269195041163359957693218431, −10.90654355827872297482482761973, −8.228086845390080192773476600405, −7.54527889367247078797641978556, −6.45064817478947638830467534786, −4.97493867965834776328255690590, −3.33255020608720458199724382089, −2.12416512879201328176508009709, 2.11608194739238174434185996990, 3.89762355216051094627785199579, 4.63889110241892541940235687206, 5.7414785725942689593431267215, 7.847813569541507806710771697255, 8.96031140042146818164921751421, 10.71418942157101797343277020207, 11.36157169621747228063100482471, 12.70975286403331488191126234851, 13.97498149660048497670573796191, 15.00798644508859618055574919333, 15.911954442935816000774531925343, 16.74641283621266643446728902844, 18.71694395542793841583243019343, 20.11343963756053627135711820985, 20.829670171902801572337805497996, 21.4071137672641415987828002259, 22.718770706455180997684646466472, 23.82821789446622455767901094093, 24.52192500462472943301609965273, 25.94995067979923822628162380188, 27.16716201149786016375868918384, 28.20071492091146746929167942120, 28.855616850341338047637022983473, 30.56147903294024453094208453166

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