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Properties

Label 1-967-967.124-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: $\chi_{967} (124, \cdot )$
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

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

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