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Properties

Label 1-97-97.77-r1-0-0
Degree $1$
Conductor $97$
Sign $0.767 - 0.640i$
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  + (−0.923 + 0.382i)2-s + (−0.382 + 0.923i)3-s + (0.707 − 0.707i)4-s + (0.195 + 0.980i)5-s − i·6-s + (0.195 − 0.980i)7-s + (−0.382 + 0.923i)8-s + (−0.707 − 0.707i)9-s + (−0.555 − 0.831i)10-s + (0.382 − 0.923i)11-s + (0.382 + 0.923i)12-s + (−0.980 + 0.195i)13-s + (0.195 + 0.980i)14-s + (−0.980 − 0.195i)15-s − i·16-s + (−0.980 + 0.195i)17-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)2-s + (−0.382 + 0.923i)3-s + (0.707 − 0.707i)4-s + (0.195 + 0.980i)5-s − i·6-s + (0.195 − 0.980i)7-s + (−0.382 + 0.923i)8-s + (−0.707 − 0.707i)9-s + (−0.555 − 0.831i)10-s + (0.382 − 0.923i)11-s + (0.382 + 0.923i)12-s + (−0.980 + 0.195i)13-s + (0.195 + 0.980i)14-s + (−0.980 − 0.195i)15-s − i·16-s + (−0.980 + 0.195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.767 - 0.640i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.767 - 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5484279878 - 0.1986804735i\)
\(L(\frac12)\) \(\approx\) \(0.5484279878 - 0.1986804735i\)
\(L(1)\) \(\approx\) \(0.5664984035 + 0.1356318603i\)
\(L(1)\) \(\approx\) \(0.5664984035 + 0.1356318603i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.923 + 0.382i)T \)
3 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (0.195 + 0.980i)T \)
7 \( 1 + (0.195 - 0.980i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (-0.980 + 0.195i)T \)
17 \( 1 + (-0.980 + 0.195i)T \)
19 \( 1 + (-0.195 - 0.980i)T \)
23 \( 1 + (0.555 - 0.831i)T \)
29 \( 1 + (0.555 - 0.831i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (0.831 - 0.555i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.382 + 0.923i)T \)
59 \( 1 + (-0.555 - 0.831i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.980 + 0.195i)T \)
71 \( 1 + (-0.831 + 0.555i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (-0.195 - 0.980i)T \)
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

−29.54056750981098103919675952998, −28.870479063067337087251207070007, −28.0618631875830571952232013729, −27.24187081927452773363089447376, −25.38914117822416242664470004454, −24.994803630203881311635652114324, −24.115291486407840352363102573264, −22.50452593343806727292796165160, −21.35453537771808255267128907664, −20.0978783764473819601604511346, −19.35779365699155255368224044176, −18.07387512127136836533986624211, −17.450547264973118183327324654939, −16.47465480022747568302965513936, −15.05216490245258965205872904846, −13.10959949091901009128673894501, −12.24917669321707497327667332544, −11.61634910995233067570014713656, −9.8625416769169325201402729614, −8.76862886355002092940885395905, −7.74977043643529361796918525108, −6.41809610482027274186585129236, −4.92340232135940723667117613902, −2.42078340491806252461064175766, −1.38690856108856698407691299988, 0.37926829059784614000341983268, 2.74208514188709039770229416514, 4.52320794204955816836664285861, 6.19896609681969638423705967268, 7.089769761444108534960574190933, 8.70459630429188317263034166409, 9.9299992930024951799862336596, 10.781973429406940398450455490268, 11.47053384732146704648029646609, 13.947646408294717178733165641424, 14.84323029243111221923001193370, 15.88638341275151109438981756283, 17.13447433758943222565694895090, 17.551277160506339527068393432319, 19.096835809016994089337424691439, 20.00151780840254769600464303694, 21.32467796864765197708225809265, 22.34927339701372092249538887378, 23.50258742815677095135908313337, 24.6210871554925777770200127184, 26.12824257508922291867983603152, 26.77062218281186084059253004203, 27.133969246080949993464060668580, 28.63492703896029901273745637420, 29.40579775996159937974922851161

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