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Properties

Label 1-801-801.671-r0-0-0
Degree $1$
Conductor $801$
Sign $-0.105 - 0.994i$
Analytic cond. $3.71982$
Root an. cond. $3.71982$
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  + (−0.580 − 0.814i)2-s + (−0.327 + 0.945i)4-s + (0.690 − 0.723i)5-s + (0.520 + 0.853i)7-s + (0.959 − 0.281i)8-s + (−0.989 − 0.142i)10-s + (0.723 − 0.690i)11-s + (−0.560 − 0.828i)13-s + (0.393 − 0.919i)14-s + (−0.786 − 0.618i)16-s + (0.909 − 0.415i)17-s + (0.599 − 0.800i)19-s + (0.458 + 0.888i)20-s + (−0.981 − 0.189i)22-s + (−0.118 + 0.992i)23-s + ⋯
L(s)  = 1  + (−0.580 − 0.814i)2-s + (−0.327 + 0.945i)4-s + (0.690 − 0.723i)5-s + (0.520 + 0.853i)7-s + (0.959 − 0.281i)8-s + (−0.989 − 0.142i)10-s + (0.723 − 0.690i)11-s + (−0.560 − 0.828i)13-s + (0.393 − 0.919i)14-s + (−0.786 − 0.618i)16-s + (0.909 − 0.415i)17-s + (0.599 − 0.800i)19-s + (0.458 + 0.888i)20-s + (−0.981 − 0.189i)22-s + (−0.118 + 0.992i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 801 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 801 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(801\)    =    \(3^{2} \cdot 89\)
Sign: $-0.105 - 0.994i$
Analytic conductor: \(3.71982\)
Root analytic conductor: \(3.71982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{801} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 801,\ (0:\ ),\ -0.105 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8676707495 - 0.9641578372i\)
\(L(\frac12)\) \(\approx\) \(0.8676707495 - 0.9641578372i\)
\(L(1)\) \(\approx\) \(0.8658445252 - 0.4654804065i\)
\(L(1)\) \(\approx\) \(0.8658445252 - 0.4654804065i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.580 - 0.814i)T \)
5 \( 1 + (0.690 - 0.723i)T \)
7 \( 1 + (0.520 + 0.853i)T \)
11 \( 1 + (0.723 - 0.690i)T \)
13 \( 1 + (-0.560 - 0.828i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (0.599 - 0.800i)T \)
23 \( 1 + (-0.118 + 0.992i)T \)
29 \( 1 + (-0.853 + 0.520i)T \)
31 \( 1 + (-0.919 - 0.393i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.436 - 0.899i)T \)
43 \( 1 + (0.0237 - 0.999i)T \)
47 \( 1 + (-0.189 - 0.981i)T \)
53 \( 1 + (0.755 + 0.654i)T \)
59 \( 1 + (0.899 - 0.436i)T \)
61 \( 1 + (-0.304 + 0.952i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (-0.281 + 0.959i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (-0.618 + 0.786i)T \)
83 \( 1 + (0.771 - 0.636i)T \)
97 \( 1 + (0.235 - 0.971i)T \)
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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

−22.74230834044526319241903636717, −21.80202736942298952222274292315, −20.78975574640533498518396490126, −19.93196539960462755699623532861, −19.061432308689145913034839289919, −18.27308300006272921638666822351, −17.625572120650563222778587820156, −16.77509188649439108618469017474, −16.44480254700546320681589356115, −14.79769127951828715338017640341, −14.570355388978119685674278435921, −14.01314786418529600826502616322, −12.868736782465570859501578945678, −11.56961535333888747249465556512, −10.69210232354804937703159348687, −9.83069609399578865448348984136, −9.43764701604104556798164035124, −8.06659853114566516211661592285, −7.32953711216678419467222330715, −6.66011696896996567406344529342, −5.78272333571603750538094418593, −4.718117948048633362618142252063, −3.76202473610720645312884116413, −2.06321564786212600580494079038, −1.302792790370114210780622202892, 0.823105682474735434369421201785, 1.78163151895163158945948245870, 2.76082633057057758479674524524, 3.77065394032858897841835439989, 5.18973516956926151826545047180, 5.56990255679575889356529682769, 7.19316514525681142311850020743, 8.127426558683712863435498691033, 9.00431411853233719381278058747, 9.45060590187298855125924335715, 10.398012511755432543929697796643, 11.55847617764306036526376685384, 11.97089192668219728205132582044, 12.94326687252395640842253306920, 13.657116209578701627019764749708, 14.62808015974953695016391169982, 15.77955616912937687768888564028, 16.76217072234530775474292606622, 17.29190903299420582423130917533, 18.13937653803524388049523277066, 18.74478109895161537200664945607, 19.82107814291404209683043294223, 20.37002597556265739191421423559, 21.20675037479538592850078516560, 21.98538488512780594547012045827

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