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Properties

Label 1-532-532.11-r1-0-0
Degree $1$
Conductor $532$
Sign $0.781 + 0.623i$
Analytic cond. $57.1713$
Root an. cond. $57.1713$
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  + (0.5 − 0.866i)3-s + 5-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)23-s + 25-s − 27-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + 33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + 5-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)23-s + 25-s − 27-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + 33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.781 + 0.623i$
Analytic conductor: \(57.1713\)
Root analytic conductor: \(57.1713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (1:\ ),\ 0.781 + 0.623i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.439189611 + 0.8533428757i\)
\(L(\frac12)\) \(\approx\) \(2.439189611 + 0.8533428757i\)
\(L(1)\) \(\approx\) \(1.440270922 - 0.05857578032i\)
\(L(1)\) \(\approx\) \(1.440270922 - 0.05857578032i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 - T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−22.638534566716514851687259809181, −22.380107782388498674296846482711, −21.40491305614533294224315862278, −20.77243539598097119140646754824, −20.00434199468430716265988825000, −19.057786247821169976697181987235, −18.057375146431507492450792060977, −17.0268280293012719385493650351, −16.52432841309502355926954419931, −15.38217834834433250342994137006, −14.669874399340340395236071791423, −13.74827302078840895364311145165, −13.2330770975311598036016552700, −11.82437747021072936208910234744, −10.764790911172828321895846891553, −10.08206656786921265285638270299, −9.18165647711022412464092302499, −8.56540421385813687074767311637, −7.313159013164994863781890091473, −6.01964244209435853603134180735, −5.26108701885709374187508098479, −4.216788033588185650852422868826, −2.98622203908847330787265312797, −2.26429272468925924417273478847, −0.57323072538265721606276851003, 1.40387593148999245843555419322, 1.93438574052111003523990340327, 3.088056689791857662455842651757, 4.43563617040277066519966311485, 5.66244239291251933944598942924, 6.725479104923375821867773238563, 7.182304439873513700625465787003, 8.58748604124077805348049916823, 9.254532943460035067864372046942, 10.10499900783202698982164193840, 11.38385520243796751563285617236, 12.41552433757921003406476294212, 13.018953082834851046199209962573, 14.013680564962847717440872290106, 14.51946929177255820524783034424, 15.490905910443229522407526505224, 17.06091833685603631777407786683, 17.36801663294645525409804152708, 18.277130488586763624603915269103, 19.18762500670193880012861711092, 19.8882444706106581574866699570, 20.78810605393711295723221127223, 21.62926511495853156415805004765, 22.47193379547066138196857716195, 23.54755316599644472216383096734

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