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Properties

Label 1-319-319.171-r0-0-0
Degree $1$
Conductor $319$
Sign $-0.289 - 0.957i$
Analytic cond. $1.48142$
Root an. cond. $1.48142$
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  + (−0.880 − 0.473i)2-s + (−0.834 − 0.550i)3-s + (0.550 + 0.834i)4-s + (−0.983 − 0.178i)5-s + (0.473 + 0.880i)6-s + (−0.936 + 0.351i)7-s + (−0.0896 − 0.995i)8-s + (0.393 + 0.919i)9-s + (0.781 + 0.623i)10-s − i·12-s + (0.753 + 0.657i)13-s + (0.990 + 0.134i)14-s + (0.722 + 0.691i)15-s + (−0.393 + 0.919i)16-s + (−0.587 + 0.809i)17-s + (0.0896 − 0.995i)18-s + ⋯
L(s)  = 1  + (−0.880 − 0.473i)2-s + (−0.834 − 0.550i)3-s + (0.550 + 0.834i)4-s + (−0.983 − 0.178i)5-s + (0.473 + 0.880i)6-s + (−0.936 + 0.351i)7-s + (−0.0896 − 0.995i)8-s + (0.393 + 0.919i)9-s + (0.781 + 0.623i)10-s − i·12-s + (0.753 + 0.657i)13-s + (0.990 + 0.134i)14-s + (0.722 + 0.691i)15-s + (−0.393 + 0.919i)16-s + (−0.587 + 0.809i)17-s + (0.0896 − 0.995i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(1.48142\)
Root analytic conductor: \(1.48142\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (0:\ ),\ -0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1851185357 - 0.2494942141i\)
\(L(\frac12)\) \(\approx\) \(0.1851185357 - 0.2494942141i\)
\(L(1)\) \(\approx\) \(0.3811982906 - 0.1418099143i\)
\(L(1)\) \(\approx\) \(0.3811982906 - 0.1418099143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.880 - 0.473i)T \)
3 \( 1 + (-0.834 - 0.550i)T \)
5 \( 1 + (-0.983 - 0.178i)T \)
7 \( 1 + (-0.936 + 0.351i)T \)
13 \( 1 + (0.753 + 0.657i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.351 - 0.936i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (0.880 + 0.473i)T \)
37 \( 1 + (0.512 - 0.858i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (-0.512 - 0.858i)T \)
53 \( 1 + (0.473 - 0.880i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.998 + 0.0448i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + (0.393 - 0.919i)T \)
73 \( 1 + (-0.722 - 0.691i)T \)
79 \( 1 + (-0.919 + 0.393i)T \)
83 \( 1 + (0.963 - 0.266i)T \)
89 \( 1 + (-0.433 - 0.900i)T \)
97 \( 1 + (0.998 + 0.0448i)T \)
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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

−25.68392249728663778873922425017, −24.509811680099733776409895657675, −23.48343738328638921515029380811, −22.95562239386160409752314868267, −22.21746506682864985997897121595, −20.54754290419922780540280691524, −20.09098040303207055412941942502, −18.86921943422028215763360560823, −18.240846421931287481477842248393, −17.161334177042522898493325340075, −16.22462201044483186026772159213, −15.80545500273976100629335984068, −15.07841536608454957142336681279, −13.63019014052057864462094136769, −12.1387339046153991294720711011, −11.3942158738363132844705076670, −10.40404879337376744052962662487, −9.778328405282804955279681524037, −8.553325721952438571708655877888, −7.45257228468068755959081284554, −6.52714705002593862883578107872, −5.663274608314923842470424170305, −4.272551990483358204454671368738, −3.13179881410007586908090887274, −0.899123734282260299277577385670, 0.411903932632879845573360756744, 1.88856565786682972086661668860, 3.32043476626717861016853063528, 4.500903638639459740805347937094, 6.27722815488221970668451486050, 6.87419741400716236033499385161, 8.06153289638971849781043755589, 8.89126310310792113350284939449, 10.14578099461702033595110671916, 11.16772299857637619566701670845, 11.80242391294496349440429274587, 12.63438342994104450265097463958, 13.406721986951264819606292223588, 15.42155611973267433725982404733, 16.13084462222320843335487693852, 16.72500040034241487365469669135, 17.920835323674628311662344155337, 18.5963303079925856351250936407, 19.50109353437525490121674369189, 19.8939968757461241885083602317, 21.389666035207480523349886783455, 22.1877999876274970537070639915, 23.10633340765143516962104328321, 24.040444565855750595449252637409, 24.854293920843292957282159312014

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