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Properties

Label 1-389-389.58-r0-0-0
Degree $1$
Conductor $389$
Sign $-0.106 - 0.994i$
Analytic cond. $1.80650$
Root an. cond. $1.80650$
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  + (0.712 + 0.701i)2-s + (−0.986 − 0.161i)3-s + (0.0161 + 0.999i)4-s + (−0.777 + 0.628i)5-s + (−0.590 − 0.807i)6-s + (−0.884 − 0.466i)7-s + (−0.689 + 0.724i)8-s + (0.948 + 0.318i)9-s + (−0.995 − 0.0970i)10-s + (−0.177 + 0.984i)11-s + (0.145 − 0.989i)12-s + (0.966 − 0.256i)13-s + (−0.302 − 0.953i)14-s + (0.868 − 0.495i)15-s + (−0.999 + 0.0323i)16-s + (−0.995 − 0.0970i)17-s + ⋯
L(s)  = 1  + (0.712 + 0.701i)2-s + (−0.986 − 0.161i)3-s + (0.0161 + 0.999i)4-s + (−0.777 + 0.628i)5-s + (−0.590 − 0.807i)6-s + (−0.884 − 0.466i)7-s + (−0.689 + 0.724i)8-s + (0.948 + 0.318i)9-s + (−0.995 − 0.0970i)10-s + (−0.177 + 0.984i)11-s + (0.145 − 0.989i)12-s + (0.966 − 0.256i)13-s + (−0.302 − 0.953i)14-s + (0.868 − 0.495i)15-s + (−0.999 + 0.0323i)16-s + (−0.995 − 0.0970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $-0.106 - 0.994i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ -0.106 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009425564287 + 0.01048516429i\)
\(L(\frac12)\) \(\approx\) \(0.009425564287 + 0.01048516429i\)
\(L(1)\) \(\approx\) \(0.5843129804 + 0.3309592245i\)
\(L(1)\) \(\approx\) \(0.5843129804 + 0.3309592245i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (0.712 + 0.701i)T \)
3 \( 1 + (-0.986 - 0.161i)T \)
5 \( 1 + (-0.777 + 0.628i)T \)
7 \( 1 + (-0.884 - 0.466i)T \)
11 \( 1 + (-0.177 + 0.984i)T \)
13 \( 1 + (0.966 - 0.256i)T \)
17 \( 1 + (-0.995 - 0.0970i)T \)
19 \( 1 + (-0.852 + 0.523i)T \)
23 \( 1 + (-0.777 - 0.628i)T \)
29 \( 1 + (0.393 - 0.919i)T \)
31 \( 1 + (-0.0485 - 0.998i)T \)
37 \( 1 + (0.797 - 0.603i)T \)
41 \( 1 + (-0.852 - 0.523i)T \)
43 \( 1 + (0.898 + 0.438i)T \)
47 \( 1 + (-0.852 - 0.523i)T \)
53 \( 1 + (-0.816 - 0.577i)T \)
59 \( 1 + (-0.423 + 0.905i)T \)
61 \( 1 + (0.966 - 0.256i)T \)
67 \( 1 + (-0.363 + 0.931i)T \)
71 \( 1 + (0.0161 + 0.999i)T \)
73 \( 1 + (-0.999 - 0.0323i)T \)
79 \( 1 + (-0.0485 + 0.998i)T \)
83 \( 1 + (-0.363 - 0.931i)T \)
89 \( 1 + (-0.937 - 0.348i)T \)
97 \( 1 + (0.0161 - 0.999i)T \)
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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

−23.72487984249456211077476825369, −22.99376124043954251271681441828, −21.923152162580200741591170187528, −21.60142652271089709397845829017, −20.43349695956199761530135630068, −19.48051906595393740811762084154, −18.83510806162125883658336013370, −17.849294472243894267104235245769, −16.36954225693436596181492509784, −15.91890567185696632804872780689, −15.22566334728782892062141412368, −13.59903002892937529672818963548, −12.9082227984383852437483630326, −12.15933750512924730119277888611, −11.26262459654552920006038182629, −10.731716671340520277661014992748, −9.412610661986904763265524466084, −8.54283118229831458352013611788, −6.67779872055132653097402478997, −6.04736516857963763182063963822, −4.992472905499899884564596234737, −4.06127046804486629421892498940, −3.15483915546651150195526239793, −1.40514956982287758139790979455, −0.00763350688065557574153224011, 2.42488520755997335791032917200, 3.98298911356608034765730627372, 4.344303759141769440447621836316, 5.95388321686890869784678706253, 6.54959924462972741175200660930, 7.31317287725722866201921407573, 8.2809786381059636201906113354, 9.95697835811165408603596286082, 10.92104677312500729598309073020, 11.8060432956698645232970576264, 12.75545944926615966179197447666, 13.3167787775511291595504916052, 14.63767773077619872739626855236, 15.66153296847058457230776442306, 16.01422983338787685467820109523, 17.06127715286500054416192461543, 17.92550515504446364503314905089, 18.74587651124636362215161952112, 19.95882858190219322192384702454, 20.96014088417148025638617242320, 22.269939961662565001661535501350, 22.63832350271563001003987820445, 23.28801472970291822186438341141, 23.83898333653703540804744577959, 24.99615367385212210568017960058

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