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Properties

Label 1-319-319.173-r1-0-0
Degree $1$
Conductor $319$
Sign $0.970 - 0.242i$
Analytic cond. $34.2813$
Root an. cond. $34.2813$
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  + (0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (−0.309 − 0.951i)6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s − 12-s + (−0.309 + 0.951i)13-s + (0.809 − 0.587i)14-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.809 − 0.587i)18-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (−0.309 − 0.951i)6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s − 12-s + (−0.309 + 0.951i)13-s + (0.809 − 0.587i)14-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.809 − 0.587i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(34.2813\)
Root analytic conductor: \(34.2813\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (1:\ ),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.975812655 - 0.3658061893i\)
\(L(\frac12)\) \(\approx\) \(2.975812655 - 0.3658061893i\)
\(L(1)\) \(\approx\) \(1.585550907 - 0.5275101745i\)
\(L(1)\) \(\approx\) \(1.585550907 - 0.5275101745i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−25.00018560080073000000176230328, −24.306406343194257486165057012, −23.392626856524677192394155702468, −22.2987797747203613578158649618, −21.28904085569709369363347263663, −20.70433788863753874033022238263, −19.88441200151556189827417505339, −18.508539427728648530760606009489, −17.29066905771034104004073952698, −16.84943600715721988968165194505, −15.74258844098168473458484634828, −15.013150509723726162574867132924, −14.0876393655680954639106170840, −13.36679869959395757050195681919, −12.530041709766043236432171576451, −10.9364926901540191355962345337, −9.66636548552877858481562000924, −8.90031573341954057788673883268, −8.00135781345368706552250618979, −7.28435294987115872374017086993, −5.547394137932033107170402708168, −4.80439807361477790875663226978, −4.03256451482309436320013143031, −2.58755174232313934436942576403, −0.73553157410178355830529433022, 1.50543301574464552436575473878, 2.195530317694804362318169512985, 3.19518970913692417604369423107, 4.32599980553441444912526757487, 5.78245112027034053327127762257, 6.853058092233154513142582873202, 8.20181478924724043863194536269, 9.040794128422663353874912289876, 10.10289509690587626039836255395, 11.117013288041133460591511119263, 12.0386931811889496066025056807, 12.93535174945673665263176478249, 13.98917758257257759494122148251, 14.67018353725651037384893309972, 15.08990875531267449428940106538, 17.2048181244766172643515620126, 18.11717662501245453976940870396, 18.91361086703762068659227453703, 19.262527712428900111134783211180, 20.52480665296935016954944766415, 21.442391633503382261712779359624, 21.75656437964906827838199653268, 23.211859384497717395086702592458, 23.809300649567949419639199788738, 24.89170904454863970285847357285

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