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Properties

Label 1-967-967.437-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.266 + 0.963i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (−0.0812 − 0.996i)2-s + (−0.844 − 0.536i)3-s + (−0.986 + 0.161i)4-s + (−0.889 + 0.457i)5-s + (−0.465 + 0.884i)6-s + (−0.922 + 0.386i)7-s + (0.241 + 0.970i)8-s + (0.425 + 0.905i)9-s + (0.527 + 0.849i)10-s + (0.787 − 0.615i)11-s + (0.919 + 0.392i)12-s + (0.139 − 0.990i)13-s + (0.460 + 0.887i)14-s + (0.995 + 0.0909i)15-s + (0.947 − 0.319i)16-s + (−0.668 + 0.744i)17-s + ⋯
L(s)  = 1  + (−0.0812 − 0.996i)2-s + (−0.844 − 0.536i)3-s + (−0.986 + 0.161i)4-s + (−0.889 + 0.457i)5-s + (−0.465 + 0.884i)6-s + (−0.922 + 0.386i)7-s + (0.241 + 0.970i)8-s + (0.425 + 0.905i)9-s + (0.527 + 0.849i)10-s + (0.787 − 0.615i)11-s + (0.919 + 0.392i)12-s + (0.139 − 0.990i)13-s + (0.460 + 0.887i)14-s + (0.995 + 0.0909i)15-s + (0.947 − 0.319i)16-s + (−0.668 + 0.744i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.266 + 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03937070519 - 0.05171196734i\)
\(L(\frac12)\) \(\approx\) \(-0.03937070519 - 0.05171196734i\)
\(L(1)\) \(\approx\) \(0.4085457613 - 0.2626654974i\)
\(L(1)\) \(\approx\) \(0.4085457613 - 0.2626654974i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.0812 - 0.996i)T \)
3 \( 1 + (-0.844 - 0.536i)T \)
5 \( 1 + (-0.889 + 0.457i)T \)
7 \( 1 + (-0.922 + 0.386i)T \)
11 \( 1 + (0.787 - 0.615i)T \)
13 \( 1 + (0.139 - 0.990i)T \)
17 \( 1 + (-0.668 + 0.744i)T \)
19 \( 1 + (0.190 - 0.981i)T \)
23 \( 1 + (0.0876 + 0.996i)T \)
29 \( 1 + (-0.442 - 0.896i)T \)
31 \( 1 + (0.516 + 0.856i)T \)
37 \( 1 + (0.861 - 0.508i)T \)
41 \( 1 + (-0.917 - 0.398i)T \)
43 \( 1 + (-0.791 + 0.610i)T \)
47 \( 1 + (0.692 - 0.721i)T \)
53 \( 1 + (0.613 - 0.789i)T \)
59 \( 1 + (-0.235 + 0.971i)T \)
61 \( 1 + (0.113 + 0.993i)T \)
67 \( 1 + (-0.999 - 0.0195i)T \)
71 \( 1 + (-0.822 - 0.568i)T \)
73 \( 1 + (0.613 + 0.789i)T \)
79 \( 1 + (-0.209 - 0.977i)T \)
83 \( 1 + (-0.759 - 0.651i)T \)
89 \( 1 + (0.483 + 0.875i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−22.421064766363030199419661464643, −22.06335525533566376801447464803, −20.633394811799976608244597413869, −20.00577957385595064992883118684, −18.8491875633417312254356611720, −18.38102670447076237650665344234, −17.04315162670912083979252655882, −16.720232307519600670780374233382, −16.16912496398987520349688288429, −15.41878058838459504652759710879, −14.68489673577725876622047568324, −13.64833236581041733750961095191, −12.60823018779664869145638135130, −12.03790241521494604455920354860, −11.05765168709738766588712516150, −9.8913567988089597318477741918, −9.36635498419322939616433191058, −8.52842692696943323951280585869, −7.219863703960424300080891588512, −6.75181803309970628291082284946, −5.9438134966051934331856825183, −4.674583289711770395083510166396, −4.29134403194972519316635369966, −3.47035683557227962558485408975, −1.194247647144947468660486818376, 0.0440355894580822102687198700, 1.099226238196884493799659668981, 2.49276640522339501021764754850, 3.368430899670378924700116084766, 4.193940379079836259971597870928, 5.39963058540333697061329886857, 6.26204015823839903620096004409, 7.194286956671376625882613112380, 8.213793378038124809980735504439, 9.04644031616569432871707726305, 10.20620858085910436357769893362, 10.85226098810009775909728650734, 11.70875648680912518074187200498, 12.02897053682889250499227326129, 13.22144533624737645364626732911, 13.38989733699775436742497163029, 14.907190133001628373889737859609, 15.68339085327064877587221358785, 16.63914988362023901976315804585, 17.52983242686675919861058299435, 18.17670604249476443839179588756, 19.03919912902810356909971092587, 19.59438277282378503627839870853, 19.9256308918112632721922151087, 21.52630820525823575962898432821

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