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Properties

Label 1-97-97.30-r1-0-0
Degree $1$
Conductor $97$
Sign $-0.691 + 0.722i$
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  + (0.382 + 0.923i)2-s + (−0.923 − 0.382i)3-s + (−0.707 + 0.707i)4-s + (0.831 + 0.555i)5-s − i·6-s + (0.831 − 0.555i)7-s + (−0.923 − 0.382i)8-s + (0.707 + 0.707i)9-s + (−0.195 + 0.980i)10-s + (0.923 + 0.382i)11-s + (0.923 − 0.382i)12-s + (−0.555 + 0.831i)13-s + (0.831 + 0.555i)14-s + (−0.555 − 0.831i)15-s − i·16-s + (−0.555 + 0.831i)17-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.923 − 0.382i)3-s + (−0.707 + 0.707i)4-s + (0.831 + 0.555i)5-s − i·6-s + (0.831 − 0.555i)7-s + (−0.923 − 0.382i)8-s + (0.707 + 0.707i)9-s + (−0.195 + 0.980i)10-s + (0.923 + 0.382i)11-s + (0.923 − 0.382i)12-s + (−0.555 + 0.831i)13-s + (0.831 + 0.555i)14-s + (−0.555 − 0.831i)15-s − i·16-s + (−0.555 + 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ -0.691 + 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6086008077 + 1.425740799i\)
\(L(\frac12)\) \(\approx\) \(0.6086008077 + 1.425740799i\)
\(L(1)\) \(\approx\) \(0.8870100423 + 0.6686603712i\)
\(L(1)\) \(\approx\) \(0.8870100423 + 0.6686603712i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (0.382 + 0.923i)T \)
3 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 + (0.831 + 0.555i)T \)
7 \( 1 + (0.831 - 0.555i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 + (-0.555 + 0.831i)T \)
17 \( 1 + (-0.555 + 0.831i)T \)
19 \( 1 + (-0.831 - 0.555i)T \)
23 \( 1 + (0.195 + 0.980i)T \)
29 \( 1 + (0.195 + 0.980i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
37 \( 1 + (-0.980 - 0.195i)T \)
41 \( 1 + (0.980 - 0.195i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (0.923 - 0.382i)T \)
59 \( 1 + (-0.195 + 0.980i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.555 + 0.831i)T \)
71 \( 1 + (0.980 + 0.195i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.831 - 0.555i)T \)
89 \( 1 + (0.923 + 0.382i)T \)
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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

−29.48406958179631613498992331773, −28.46529813479297390658569843034, −27.67892945708414281906633694160, −26.96511280334682524085770645959, −24.833494492935833671467959362036, −24.21990518545733217942320411298, −22.71965464298561476891891842846, −22.03217211365125005093935328295, −21.1274337520767795804976302059, −20.36725990943556308468918939087, −18.73319322116276110658059670060, −17.69753532576887060221827076088, −16.9480878770674192208890694827, −15.25197785403910595761117251694, −14.14537489181945716317991510330, −12.75165521890385923932957178573, −11.9134556166032867433383709356, −10.86412032558693357171708039630, −9.753105412582440752448195790426, −8.70397275365782962477010841803, −6.18894146420669907248910024317, −5.25096154335157294816944046047, −4.2863203871060639910658048605, −2.222077807689065151360441341340, −0.72185602346934564771666374972, 1.72204063813781357952877903506, 4.197998222469448900387601206607, 5.32352820485880952303234366069, 6.65515665782547015819276542057, 7.1754634504260827658704209177, 8.95653211635455138183288884644, 10.501994714433026295872757048312, 11.74632936226416115393509058706, 13.040087842924821092273241533018, 14.09470081099802042535156712638, 14.96532179877829142294921674448, 16.61980832571858420455382233872, 17.45348974702473481001637269732, 17.843811188590661508663446714086, 19.35973288001056513616053459569, 21.47129487657514236608330329575, 21.88050206098978662260422883317, 23.09762913901626800669662862703, 23.94957717000854766361256828356, 24.7965445000924286938847030664, 25.91105615870423809188566948368, 27.0186297701899009960513824478, 28.00138922777496658441357042626, 29.48740337734195616739928922818, 30.2189637164987893716821950809

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