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Properties

Label 1-1449-1449.1222-r0-0-0
Degree $1$
Conductor $1449$
Sign $-0.979 - 0.199i$
Analytic cond. $6.72912$
Root an. cond. $6.72912$
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  + (−0.959 + 0.281i)2-s + (0.841 − 0.540i)4-s + (−0.786 + 0.618i)5-s + (−0.654 + 0.755i)8-s + (0.580 − 0.814i)10-s + (0.723 − 0.690i)11-s + (−0.995 − 0.0950i)13-s + (0.415 − 0.909i)16-s + (0.0475 − 0.998i)17-s + (0.0475 + 0.998i)19-s + (−0.327 + 0.945i)20-s + (−0.5 + 0.866i)22-s + (0.235 − 0.971i)25-s + (0.981 − 0.189i)26-s + (0.0475 − 0.998i)29-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (0.841 − 0.540i)4-s + (−0.786 + 0.618i)5-s + (−0.654 + 0.755i)8-s + (0.580 − 0.814i)10-s + (0.723 − 0.690i)11-s + (−0.995 − 0.0950i)13-s + (0.415 − 0.909i)16-s + (0.0475 − 0.998i)17-s + (0.0475 + 0.998i)19-s + (−0.327 + 0.945i)20-s + (−0.5 + 0.866i)22-s + (0.235 − 0.971i)25-s + (0.981 − 0.189i)26-s + (0.0475 − 0.998i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $-0.979 - 0.199i$
Analytic conductor: \(6.72912\)
Root analytic conductor: \(6.72912\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (1222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1449,\ (0:\ ),\ -0.979 - 0.199i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005955563154 + 0.05896683159i\)
\(L(\frac12)\) \(\approx\) \(0.005955563154 + 0.05896683159i\)
\(L(1)\) \(\approx\) \(0.5063119414 + 0.08146679820i\)
\(L(1)\) \(\approx\) \(0.5063119414 + 0.08146679820i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.959 + 0.281i)T \)
5 \( 1 + (-0.786 + 0.618i)T \)
11 \( 1 + (0.723 - 0.690i)T \)
13 \( 1 + (-0.995 - 0.0950i)T \)
17 \( 1 + (0.0475 - 0.998i)T \)
19 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (0.0475 - 0.998i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (-0.786 - 0.618i)T \)
41 \( 1 + (0.928 + 0.371i)T \)
43 \( 1 + (-0.327 + 0.945i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.995 + 0.0950i)T \)
59 \( 1 + (0.415 + 0.909i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.888 - 0.458i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (0.928 - 0.371i)T \)
89 \( 1 + (0.981 - 0.189i)T \)
97 \( 1 + (-0.786 + 0.618i)T \)
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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

−20.27456423960873060040522007311, −19.35427986439364468918076228511, −19.202332023259015034511516929928, −17.97303554446772077220384508099, −17.172754598716060200286858656135, −16.87963728785183349144954564701, −15.86532286375686277944870657700, −15.21833945066218086766187157588, −14.53205968543315299412535091763, −13.09772001071970549504909616919, −12.37646354825386084439528860369, −11.927395117017117876341581286157, −11.04392454524505449969183380198, −10.260819270886342439908643433, −9.20914215728447183520346913671, −8.91532558153596971422334092984, −7.81019022124563293183040759436, −7.269774017892326782865323675189, −6.45333249700446961820316084368, −5.126734418552218354827103847978, −4.200939566609602969900519954246, −3.36916594876458137091053604823, −2.17833020396898956174428090151, −1.313092934024606056246141650653, −0.03425537938249706476374364563, 1.212196396253706784767571942098, 2.49208158792082707675319758839, 3.25124055829734506051009815738, 4.36710914128115260949792060161, 5.583292460309733028887150670950, 6.39568420956574355487907162188, 7.31133350150393308442929762250, 7.71583192560271787044617856938, 8.69711765021306490174693733153, 9.46673354861528335105580664398, 10.29696298516954146395209417052, 11.04178094606403186141623030022, 11.823560560749046103821903100902, 12.280560471750224791388894030849, 13.85725970883533827329733472089, 14.54766862032277503077585091561, 15.06692979050157830530282375283, 16.18897884440442026700903155168, 16.37349107052958565894609262357, 17.46705907019059091348507644412, 18.10123012776918612758161868275, 19.04166967755838186961473460618, 19.31009708826976655159337888556, 20.08520523602046902390721120044, 20.895111265870152701593267048211

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