Properties

Label 1-967-967.39-r0-0-0
Degree $1$
Conductor $967$
Sign $0.0778 + 0.996i$
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  + (−0.715 − 0.699i)2-s + (0.203 − 0.979i)3-s + (0.0227 + 0.999i)4-s + (0.538 + 0.842i)5-s + (−0.829 + 0.557i)6-s + (−0.419 − 0.907i)7-s + (0.682 − 0.730i)8-s + (−0.917 − 0.398i)9-s + (0.203 − 0.979i)10-s + (−0.917 − 0.398i)11-s + (0.983 + 0.181i)12-s + (0.113 − 0.993i)13-s + (−0.334 + 0.942i)14-s + (0.934 − 0.356i)15-s + (−0.998 + 0.0455i)16-s + (−0.775 + 0.631i)17-s + ⋯
L(s)  = 1  + (−0.715 − 0.699i)2-s + (0.203 − 0.979i)3-s + (0.0227 + 0.999i)4-s + (0.538 + 0.842i)5-s + (−0.829 + 0.557i)6-s + (−0.419 − 0.907i)7-s + (0.682 − 0.730i)8-s + (−0.917 − 0.398i)9-s + (0.203 − 0.979i)10-s + (−0.917 − 0.398i)11-s + (0.983 + 0.181i)12-s + (0.113 − 0.993i)13-s + (−0.334 + 0.942i)14-s + (0.934 − 0.356i)15-s + (−0.998 + 0.0455i)16-s + (−0.775 + 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0778 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0778 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.0778 + 0.996i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.0778 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005076130489 + 0.004695394750i\)
\(L(\frac12)\) \(\approx\) \(0.005076130489 + 0.004695394750i\)
\(L(1)\) \(\approx\) \(0.5017987590 - 0.3234067983i\)
\(L(1)\) \(\approx\) \(0.5017987590 - 0.3234067983i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.715 - 0.699i)T \)
3 \( 1 + (0.203 - 0.979i)T \)
5 \( 1 + (0.538 + 0.842i)T \)
7 \( 1 + (-0.419 - 0.907i)T \)
11 \( 1 + (-0.917 - 0.398i)T \)
13 \( 1 + (0.113 - 0.993i)T \)
17 \( 1 + (-0.775 + 0.631i)T \)
19 \( 1 + (-0.949 + 0.313i)T \)
23 \( 1 + (0.854 - 0.519i)T \)
29 \( 1 + (-0.917 + 0.398i)T \)
31 \( 1 + (0.291 + 0.956i)T \)
37 \( 1 + (0.113 - 0.993i)T \)
41 \( 1 + (-0.0682 - 0.997i)T \)
43 \( 1 + (-0.158 + 0.987i)T \)
47 \( 1 + (0.934 - 0.356i)T \)
53 \( 1 + (-0.877 - 0.480i)T \)
59 \( 1 + (-0.877 + 0.480i)T \)
61 \( 1 + (0.898 + 0.439i)T \)
67 \( 1 + (0.682 + 0.730i)T \)
71 \( 1 + (0.962 + 0.269i)T \)
73 \( 1 + (-0.877 + 0.480i)T \)
79 \( 1 + (-0.877 + 0.480i)T \)
83 \( 1 + (-0.829 + 0.557i)T \)
89 \( 1 + (-0.974 + 0.225i)T \)
97 \( 1 + T \)
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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

−22.24089810047444343543257752252, −21.47819319101658183616642368140, −20.70766382387962778195708079170, −20.09277593524072748693275322541, −19.026860030976263277679121650676, −18.45779582642050908490745513867, −17.21686806401370252909854664629, −16.94745119630241800125665087471, −15.86392297117448484265371836032, −15.55778369026418654087833118511, −14.792957059782457985207685123911, −13.67209671474062467297444742287, −13.04918488803923792802650751325, −11.67161077907927115739543889985, −10.87184766821081184323906524200, −9.76574317751710082392480523795, −9.34413152146191311394476273284, −8.77500487977346916353974192522, −7.94893999987751025773907022259, −6.630992340264829692008187631754, −5.78342540088300965236957418509, −4.99084137105812992077925542217, −4.38838910074792950929399625793, −2.63738520388977160188865989081, −1.90355794966134076512879204488, 0.00358384427823642650083120979, 1.271457943929868121662026067983, 2.37119070489931065309838446031, 2.99703151788074125547638403845, 3.89739577950497153617380161987, 5.60176450169719615625619663216, 6.65966900389625082322215171677, 7.22040177494318374396853687175, 8.10174760538535465621105763470, 8.843938690188628154270672933146, 10.01886341693674828150213333676, 10.84378684703007096690644185835, 11.00052775507035253750508944415, 12.69968919436226821324153434414, 12.90200916244594380318573246358, 13.66464086322519189607613874945, 14.59910545716281774718590521250, 15.69720469942264451112368490098, 16.89029015305250079718049973545, 17.47541108947784956173276433611, 18.10192037147538518892073332106, 18.86690462318033149513647641913, 19.38875329751521676066561624842, 20.22418670188656251099398956757, 20.9146493873674743119700932781

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