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Properties

Label 1-97-97.94-r0-0-0
Degree $1$
Conductor $97$
Sign $0.821 - 0.570i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.821 - 0.570i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ 0.821 - 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.377246488 - 0.4310812094i\)
\(L(\frac12)\) \(\approx\) \(1.377246488 - 0.4310812094i\)
\(L(1)\) \(\approx\) \(1.381052303 - 0.3788530653i\)
\(L(1)\) \(\approx\) \(1.381052303 - 0.3788530653i\)

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

−30.7522559882767586962314932937, −29.30245888096561150232247544716, −27.73573586268683383930881271164, −26.88436181504132222198215281695, −25.7600961085039459114248233215, −25.02039323334504123661947615481, −23.95633975153375102758421290773, −23.517374769223933536648517576036, −21.63334758263684076436388015024, −20.76081177402962979990231893905, −19.77862901841901685462312964873, −18.21769459101925395243919653642, −17.31617179925205834013434725207, −16.14469355925503540938053360579, −15.01009582746285226251724820507, −14.00331043179898737029450944579, −13.20465258141932194628336596693, −12.07899317090681071965619404375, −9.78491046632633856633801784094, −8.75775135720355697274035210653, −7.81996169848131427044820100957, −6.771571867261775152940869965011, −4.838899619609170886350462816862, −4.10405837191043756238250808924, −1.82407140395390925328961122459, 2.14206897671729868293750186036, 2.977632219708227384450269650597, 4.345845935320707847059008381827, 5.9382005688232807872320392544, 8.022896376029595698280034740, 8.930547894889097868290444381405, 10.44707763168189994246683418881, 11.010453362578717384781690747775, 12.65411159089490875217088183240, 13.80416603095501598677133652259, 14.68716052160121107556040617813, 15.4511968958228448336229827682, 17.718611185362649465350871167211, 18.61100602310195852697635942942, 19.46692138188261981475367811022, 20.55556868257075921868553137257, 21.715141567423808394512750135837, 22.00472580986380112508905301988, 23.656145435901971258081186872667, 24.79327205646183360555857934354, 26.12746708488253774590001374950, 27.01264255527527001558678143033, 27.76981487347153870987536491811, 29.21608007700997603988496499419, 30.38704848106112931428349269176

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