↑

Properties

Label 1-319-319.126-r1-0-0
Degree $1$
Conductor $319$
Sign $-0.614 + 0.788i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.178 + 0.983i)2-s + (−0.351 + 0.936i)3-s + (−0.936 + 0.351i)4-s + (−0.473 − 0.880i)5-s + (−0.983 − 0.178i)6-s + (−0.550 − 0.834i)7-s + (−0.512 − 0.858i)8-s + (−0.753 − 0.657i)9-s + (0.781 − 0.623i)10-s − i·12-s + (0.393 + 0.919i)13-s + (0.722 − 0.691i)14-s + (0.990 − 0.134i)15-s + (0.753 − 0.657i)16-s + (0.587 − 0.809i)17-s + (0.512 − 0.858i)18-s + ⋯
L(s)  = 1  + (0.178 + 0.983i)2-s + (−0.351 + 0.936i)3-s + (−0.936 + 0.351i)4-s + (−0.473 − 0.880i)5-s + (−0.983 − 0.178i)6-s + (−0.550 − 0.834i)7-s + (−0.512 − 0.858i)8-s + (−0.753 − 0.657i)9-s + (0.781 − 0.623i)10-s − i·12-s + (0.393 + 0.919i)13-s + (0.722 − 0.691i)14-s + (0.990 − 0.134i)15-s + (0.753 − 0.657i)16-s + (0.587 − 0.809i)17-s + (0.512 − 0.858i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $-0.614 + 0.788i$
Analytic conductor: \(34.2813\)
Root analytic conductor: \(34.2813\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (1:\ ),\ -0.614 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3868762274 + 0.7921773018i\)
\(L(\frac12)\) \(\approx\) \(0.3868762274 + 0.7921773018i\)
\(L(1)\) \(\approx\) \(0.6149782994 + 0.4126109056i\)
\(L(1)\) \(\approx\) \(0.6149782994 + 0.4126109056i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.178 + 0.983i)T \)
3 \( 1 + (-0.351 + 0.936i)T \)
5 \( 1 + (-0.473 - 0.880i)T \)
7 \( 1 + (-0.550 - 0.834i)T \)
13 \( 1 + (0.393 + 0.919i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.834 - 0.550i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (0.178 + 0.983i)T \)
37 \( 1 + (-0.0896 + 0.995i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (0.0896 + 0.995i)T \)
53 \( 1 + (0.983 - 0.178i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.266 - 0.963i)T \)
67 \( 1 + (0.222 + 0.974i)T \)
71 \( 1 + (-0.753 + 0.657i)T \)
73 \( 1 + (0.990 - 0.134i)T \)
79 \( 1 + (-0.657 + 0.753i)T \)
83 \( 1 + (-0.0448 - 0.998i)T \)
89 \( 1 + (0.433 - 0.900i)T \)
97 \( 1 + (-0.266 + 0.963i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.44150432290775112524368533559, −23.31297283752601547992176726613, −22.83106618054750600246877584736, −22.12453083968639281926090292093, −21.146258259430611619177608210174, −19.72141831810043875938214270752, −19.32782232846167656875913576637, −18.35008162132571471025040560042, −18.04655539070002323606065220449, −16.65990907694768370881378136519, −15.20213806391099968457425343277, −14.454728800183395042139695135262, −13.26291666835237550730824249571, −12.49804221399253172890852522504, −11.85023372876572039282502882803, −10.82978219330647449361470920758, −10.09307187494196368853677646319, −8.56384838796234574211031647830, −7.791309645042502936984108720080, −6.20938411629693503286724420954, −5.72586784244275783235988003909, −3.955025884185713741261546441643, −2.87537938248758012412296786449, −2.004351318162094317588518628850, −0.40855791542678102394272479193, 0.72356914670183543532359709714, 3.474522218256068772993823425852, 4.24906013716962540456300483764, 5.00330908292233716172912438135, 6.169425243477379982716121370135, 7.19255979560966810639160485465, 8.437114256304706800958165853, 9.259536805491681804604511712117, 10.10350858934853720104765308486, 11.48511652657387592288167008419, 12.46935539541659022618499988564, 13.592864782605661136275396048384, 14.394927157268083434272588780542, 15.733529072364826611629032248816, 16.06034796372577093684895244351, 16.86411892852052176161786252428, 17.4929879895069069072362765837, 18.92760482978250315707150364631, 20.019120220520429578526906640678, 20.97233797557410266900252940853, 21.793669238183664683196960773426, 22.88012945918783917139509315039, 23.479328857961604630371857398100, 24.070685311438025062598301440334, 25.43503474174079186430514322739

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