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Properties

Label 1-97-97.28-r1-0-0
Degree $1$
Conductor $97$
Sign $0.112 + 0.993i$
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 + (0.980 + 0.195i)5-s + i·6-s + (0.980 − 0.195i)7-s + (0.382 + 0.923i)8-s + (−0.707 + 0.707i)9-s + (0.831 + 0.555i)10-s + (−0.382 − 0.923i)11-s + (−0.382 + 0.923i)12-s + (0.195 − 0.980i)13-s + (0.980 + 0.195i)14-s + (0.195 + 0.980i)15-s + i·16-s + (0.195 − 0.980i)17-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)2-s + (0.382 + 0.923i)3-s + (0.707 + 0.707i)4-s + (0.980 + 0.195i)5-s + i·6-s + (0.980 − 0.195i)7-s + (0.382 + 0.923i)8-s + (−0.707 + 0.707i)9-s + (0.831 + 0.555i)10-s + (−0.382 − 0.923i)11-s + (−0.382 + 0.923i)12-s + (0.195 − 0.980i)13-s + (0.980 + 0.195i)14-s + (0.195 + 0.980i)15-s + i·16-s + (0.195 − 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.112 + 0.993i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.112 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.980835797 + 2.661750863i\)
\(L(\frac12)\) \(\approx\) \(2.980835797 + 2.661750863i\)
\(L(1)\) \(\approx\) \(2.106063090 + 1.193751533i\)
\(L(1)\) \(\approx\) \(2.106063090 + 1.193751533i\)

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 \( 1 + (0.980 + 0.195i)T \)
7 \( 1 + (0.980 - 0.195i)T \)
11 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 + (0.195 - 0.980i)T \)
17 \( 1 + (0.195 - 0.980i)T \)
19 \( 1 + (-0.980 - 0.195i)T \)
23 \( 1 + (-0.831 + 0.555i)T \)
29 \( 1 + (-0.831 + 0.555i)T \)
31 \( 1 + (-0.923 + 0.382i)T \)
37 \( 1 + (0.555 - 0.831i)T \)
41 \( 1 + (-0.555 - 0.831i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (0.831 + 0.555i)T \)
61 \( 1 + T \)
67 \( 1 + (0.195 - 0.980i)T \)
71 \( 1 + (-0.555 + 0.831i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (-0.980 - 0.195i)T \)
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

−29.91268743994250901329811206128, −28.761417419429080496258097062757, −28.10560767780696410921400236729, −25.964565074858696217375703273944, −25.259880608169779543952618609112, −24.089826968182005202662322486264, −23.67510685170426465632675081577, −22.13546427381257977352021473602, −21.01394227165092313766946576802, −20.45747824376785486386842741738, −19.041777871732185850312000187025, −18.065942260300867697018241868580, −16.88891154209956414512460616350, −14.90997159359676214306334502883, −14.34627535211527155049197063022, −13.19794453493343971934356759371, −12.43825703672123761913289390480, −11.20859610866081099909515406635, −9.771067050378553558055565209238, −8.232727056878134552391175683440, −6.72298092759049698915138994258, −5.68560076878287495975150265142, −4.2558037470489967281145463979, −2.17924111868267550516856221321, −1.70529206117758640916970959428, 2.285785229023347035260081532376, 3.56037114603839835822668426306, 5.07440676559678244680498158760, 5.77589563200868465432660290663, 7.64199752504461282623580195772, 8.81913017478918558840266366244, 10.491679215875468917244859305689, 11.25029995669998960498459386118, 13.11470849622339620981464100119, 14.05440909876664303800880468441, 14.752655357846972803644228485364, 15.93772332154581156788348839561, 16.96758633431920990412417466148, 18.07640137550021440817497167905, 20.08568137383058411081620618946, 20.99028765320205735009089299125, 21.58733717266181805599573515680, 22.50860796498942118582350631460, 23.82725266610040045942438695297, 24.97596426370069818965361663467, 25.728838325946029941658539433477, 26.76546496153567452798606789492, 27.848098736154156122873537047893, 29.44272876589120637000904057379, 30.13546605359060230826593640365

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