L-function 1-532-532.31-r1-0-0

• Label: 1-532-532.31-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: $0$

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 + ⋯

Functional equation

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

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:	$\chi_{532} (31, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 532,\ (1:\ ),\ -0.851 - 0.524i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)

	$p$	$F_p(T)$
bad	2	\( 1 \)
	7	\( 1 \)
	19	\( 1 \)
good	3	\( 1 - T \)
	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	\( 1 + (0.5 - 0.866i)T \)

Imaginary part of the first few zeros on the critical line

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

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