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Properties

Label 1-532-532.103-r1-0-0
Degree $1$
Conductor $532$
Sign $-0.851 + 0.524i$
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  − 3-s + (0.5 − 0.866i)5-s + 9-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)15-s − 17-s − 23-s + (−0.5 − 0.866i)25-s − 27-s + (0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯
L(s)  = 1  − 3-s + (0.5 − 0.866i)5-s + 9-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)15-s − 17-s − 23-s + (−0.5 − 0.866i)25-s − 27-s + (0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.851 + 0.524i$
Analytic conductor: \(57.1713\)
Root analytic conductor: \(57.1713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (1:\ ),\ -0.851 + 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03864182375 - 0.1363146047i\)
\(L(\frac12)\) \(\approx\) \(-0.03864182375 - 0.1363146047i\)
\(L(1)\) \(\approx\) \(0.7035188201 - 0.1565698979i\)
\(L(1)\) \(\approx\) \(0.7035188201 - 0.1565698979i\)

Euler product

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

−23.50579817255820676326522632784, −22.81639079512864634239627357710, −22.06842550983658488226535608182, −21.70079487767265679210847861268, −20.36481237771619122918194177795, −19.53917440288432000170828259304, −18.36809454361684376424099308579, −17.714471309252114601294128912763, −17.3606727719820321144370552474, −16.117971610631031017535540339708, −15.281752825885524002159740743098, −14.469735133863593969118149908272, −13.384112250239740383712661287400, −12.44172649916869394811811300448, −11.69129545617591927862840751188, −10.5660180538718387294737450035, −10.199181415402380944008214041552, −9.12243056683142124632536506531, −7.57017316790460922814002382325, −6.82422130127940847496768313114, −6.05436305997323603469817995698, −5.05624974067818938730356696073, −4.04378285964143122957318550170, −2.628914977020273903323418723726, −1.52423753565104126406182004337, 0.04336101959167780375597517767, 1.1477669704206203678719674555, 2.26680174402825117902311735620, 4.145156861899632928707317190596, 4.70415875153717199771896320988, 5.97285190864713827927489524108, 6.36823902937239367191793633891, 7.72294333428244808171414268474, 8.9100680503576484495241321529, 9.647057042365209573566451250004, 10.658333291795888439336705710587, 11.73122758910284637676117726259, 12.15937462799408436457953326325, 13.38143954896767880117467408477, 13.886670684387278636696108636465, 15.35282278906230129194634702953, 16.216354625244457740206055092720, 16.92030635213823254775812703602, 17.43089044001228689717324308843, 18.441929725326634565762692476762, 19.34511549771443530969147207785, 20.318840095608767213378716644537, 21.39066911578004882468620724957, 21.85001579665296595946177352198, 22.64010479914438022573465372578

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