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Properties

Label 1-97-97.75-r0-0-0
Degree $1$
Conductor $97$
Sign $-0.207 - 0.978i$
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  − 2-s − 3-s + 4-s − i·5-s + 6-s + i·7-s − 8-s + 9-s + i·10-s − 11-s − 12-s − i·13-s − i·14-s + i·15-s + 16-s − i·17-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − i·5-s + 6-s + i·7-s − 8-s + 9-s + i·10-s − 11-s − 12-s − i·13-s − i·14-s + i·15-s + 16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.207 - 0.978i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ -0.207 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2195118702 - 0.2709838852i\)
\(L(\frac12)\) \(\approx\) \(0.2195118702 - 0.2709838852i\)
\(L(1)\) \(\approx\) \(0.4389419288 - 0.1368795992i\)
\(L(1)\) \(\approx\) \(0.4389419288 - 0.1368795992i\)

Euler product

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

−29.841459466460202099639741131620, −29.33841813397548808193969620874, −28.352831959863692199237611531116, −27.133945159594020552310740058144, −26.50985229028967280972834013191, −25.57610105101270424110820374080, −23.805180486675836008473493099988, −23.44032075178551912016186742934, −21.88282647600143460534225258107, −20.95180207491081621870788410515, −19.43182329883066332122445842427, −18.52794653564330460394338017565, −17.66911429253218547013940794759, −16.69051075582370003037072195006, −15.77293485108482160268282807993, −14.386373554831793385303144665633, −12.68949462074814405605998111028, −11.21503131695548155823517201909, −10.66717335383177216474695952409, −9.7447273521408879814251311476, −7.7344714553961445615109965343, −6.94474657490022917005543743567, −5.80465014053441773584730361982, −3.73864225832041735256654048605, −1.72473267424681441337384180171, 0.54941516288487583237355999153, 2.45044595969309393949711640008, 5.022159286447815666668939850, 5.89231965169836041380711111927, 7.491983993131464172499974983789, 8.70967129756331932873721107279, 9.84040681941754960017621727161, 11.05776802435381329264328325408, 12.14445181807835135593646885275, 12.96425365199509138847810494123, 15.50231464188291418156173209057, 15.90902989444569432032956259251, 17.12842678425865073792955559960, 18.01228598666779850131918941853, 18.822271716536776996172143150111, 20.34482598142103982359117776175, 21.15469955831986685258645842823, 22.40790994942075057956329718740, 23.891494176626831388615122415857, 24.59782078740071861749578257500, 25.58381697602523909314146476636, 27.0956741314190079725162405451, 27.88600270816115195863810751233, 28.64218081126164029918950649374, 29.19639614232536785519507590907

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