Properties

Label 1-89-89.63-r1-0-0
Degree $1$
Conductor $89$
Sign $0.575 + 0.817i$
Analytic cond. $9.56437$
Root an. cond. $9.56437$
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  + (0.841 + 0.540i)2-s + (0.936 − 0.349i)3-s + (0.415 + 0.909i)4-s + (0.989 + 0.142i)5-s + (0.977 + 0.212i)6-s + (−0.800 + 0.599i)7-s + (−0.142 + 0.989i)8-s + (0.755 − 0.654i)9-s + (0.755 + 0.654i)10-s + (0.142 + 0.989i)11-s + (0.707 + 0.707i)12-s + (−0.349 − 0.936i)13-s + (−0.997 + 0.0713i)14-s + (0.977 − 0.212i)15-s + (−0.654 + 0.755i)16-s + (−0.540 − 0.841i)17-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (0.936 − 0.349i)3-s + (0.415 + 0.909i)4-s + (0.989 + 0.142i)5-s + (0.977 + 0.212i)6-s + (−0.800 + 0.599i)7-s + (−0.142 + 0.989i)8-s + (0.755 − 0.654i)9-s + (0.755 + 0.654i)10-s + (0.142 + 0.989i)11-s + (0.707 + 0.707i)12-s + (−0.349 − 0.936i)13-s + (−0.997 + 0.0713i)14-s + (0.977 − 0.212i)15-s + (−0.654 + 0.755i)16-s + (−0.540 − 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.575 + 0.817i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ 0.575 + 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.463697513 + 1.797968156i\)
\(L(\frac12)\) \(\approx\) \(3.463697513 + 1.797968156i\)
\(L(1)\) \(\approx\) \(2.276259578 + 0.7843541199i\)
\(L(1)\) \(\approx\) \(2.276259578 + 0.7843541199i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + (0.936 - 0.349i)T \)
5 \( 1 + (0.989 + 0.142i)T \)
7 \( 1 + (-0.800 + 0.599i)T \)
11 \( 1 + (0.142 + 0.989i)T \)
13 \( 1 + (-0.349 - 0.936i)T \)
17 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (0.997 + 0.0713i)T \)
23 \( 1 + (0.0713 - 0.997i)T \)
29 \( 1 + (-0.599 - 0.800i)T \)
31 \( 1 + (0.0713 + 0.997i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.349 - 0.936i)T \)
43 \( 1 + (-0.599 + 0.800i)T \)
47 \( 1 + (0.909 - 0.415i)T \)
53 \( 1 + (-0.909 - 0.415i)T \)
59 \( 1 + (-0.936 - 0.349i)T \)
61 \( 1 + (0.877 + 0.479i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (-0.989 + 0.142i)T \)
73 \( 1 + (0.654 - 0.755i)T \)
79 \( 1 + (-0.755 - 0.654i)T \)
83 \( 1 + (-0.977 - 0.212i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−30.02298068693498530415945669761, −29.36416248094124887944802075530, −28.35035679762145614686281022134, −26.7493193177683616763746006612, −25.87968161267210525501825884392, −24.69719419750218586837367319372, −23.86412111347439629317755548995, −22.122610364190419256864431459158, −21.693016898112957525780985218423, −20.57155263464562988021089937408, −19.64865285793402262690709605261, −18.76262602477612752501721394438, −16.80115367933412773334065223572, −15.72643354112934979280962350857, −14.27857135138546642838469972077, −13.6781232734543255657533241328, −12.859787061085113803508860051683, −11.06122304402992719774488199818, −9.87318269347149548187761437214, −9.1193709308205382350304809813, −7.01386302884470822632037020535, −5.67557616116132188326525619700, −4.10500690463073644551772715232, −3.00634602093942234076593350373, −1.57895982574319430186703516706, 2.252652378668407051182175029144, 3.15995025616911233544008203539, 5.01110327505211871421037793520, 6.40860899345282390891951849941, 7.37222286296006266923652818445, 8.91251339746665269770204163537, 9.97980022586496227060389849736, 12.21237897910845746182384737421, 13.006958094943559444087796415515, 13.94235251742844296183602099071, 14.97213783191306750871578568639, 15.876968740087843249676268317, 17.47504308041238509056356741418, 18.42165208649375848754633092739, 20.07936752969461277092393840897, 20.80294152465218837258121294995, 22.15779067808784043372123917019, 22.769682911829406046039438868137, 24.58491939968854068618916322478, 25.01103547374932288111856001229, 25.79876726308858949400348904351, 26.73208361144343568773922543912, 28.72102895435760222472719103929, 29.68451518586494478802623956882, 30.60803589162253240077969507533

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