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Properties

Label 1-532-532.335-r1-0-0
Degree $1$
Conductor $532$
Sign $-0.813 + 0.582i$
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 + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s − 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 + (−0.5 − 0.866i)25-s − 27-s + (0.5 + 0.866i)29-s − 31-s + (−0.5 + 0.866i)33-s − 37-s − 39-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s − 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 + (−0.5 − 0.866i)25-s − 27-s + (0.5 + 0.866i)29-s − 31-s + (−0.5 + 0.866i)33-s − 37-s − 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.813 + 0.582i$
Analytic conductor: \(57.1713\)
Root analytic conductor: \(57.1713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (1:\ ),\ -0.813 + 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3635165898 - 1.131922398i\)
\(L(\frac12)\) \(\approx\) \(-0.3635165898 - 1.131922398i\)
\(L(1)\) \(\approx\) \(0.8858915522 - 0.6344422651i\)
\(L(1)\) \(\approx\) \(0.8858915522 - 0.6344422651i\)

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 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 - T \)
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 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
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

−23.68228537243809159071211539630, −22.69643879998264902951533019714, −21.986965978499035517541915443653, −21.16091765774086329795545766315, −20.78171893253597779047234428435, −19.367477809853015971954006489013, −18.95930087821984378626715425726, −17.83264366674991313368537047820, −16.90912432791525108017732511140, −16.05965560966275858753467422480, −15.08399522128245996568164481513, −14.50367112782692981734860852362, −13.74571519869103350692671375616, −12.73761259946196952014995131261, −11.398932633532710689963384733890, −10.50797400351262033509409646561, −10.02059609699873892323789621241, −9.02723017876685244243513490702, −8.02200632213412731724727598944, −7.02034429596559183746589809766, −5.85792370786733437326299425146, −4.90176665873155955366521727111, −3.79255322795703829509008210119, −2.77730793472336224500808246166, −1.983505924597395711463993874993, 0.258780652773546407962305464357, 1.33171239296466862050103505078, 2.45164340007763903985395637805, 3.38257331787598000267090048654, 5.1007538036044224201796651539, 5.57020503024303049855258350567, 6.99491337539961118708070841055, 7.76715606775173429172613425258, 8.62889295273213064581263591074, 9.49307912550867847409525934754, 10.451678481220632331574388661097, 11.83201737873744616503125828750, 12.61513061341047387423991082919, 13.246202495189585725274477373996, 13.959575998795239091997685340688, 15.003501950098003186806855726748, 15.95901351845396900093459790137, 16.979419740302187557222530366867, 17.847036499022473431738837531357, 18.40645965399008065561368098660, 19.49984905050847206936989349772, 20.27981536309937113736312578274, 20.83802137684713199441438456213, 21.77493518432801468385427461049, 23.02672639254912121435210004341

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