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Properties

Label 1-967-967.114-r0-0-0
Degree $1$
Conductor $967$
Sign $0.501 - 0.865i$
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  + (0.254 − 0.967i)2-s + (−0.511 + 0.859i)3-s + (−0.870 − 0.491i)4-s + (−0.0552 + 0.998i)5-s + (0.701 + 0.712i)6-s + (−0.533 + 0.845i)7-s + (−0.696 + 0.717i)8-s + (−0.477 − 0.878i)9-s + (0.951 + 0.307i)10-s + (0.389 − 0.921i)11-s + (0.867 − 0.497i)12-s + (−0.992 − 0.123i)13-s + (0.682 + 0.730i)14-s + (−0.829 − 0.557i)15-s + (0.516 + 0.856i)16-s + (0.737 − 0.675i)17-s + ⋯
L(s)  = 1  + (0.254 − 0.967i)2-s + (−0.511 + 0.859i)3-s + (−0.870 − 0.491i)4-s + (−0.0552 + 0.998i)5-s + (0.701 + 0.712i)6-s + (−0.533 + 0.845i)7-s + (−0.696 + 0.717i)8-s + (−0.477 − 0.878i)9-s + (0.951 + 0.307i)10-s + (0.389 − 0.921i)11-s + (0.867 − 0.497i)12-s + (−0.992 − 0.123i)13-s + (0.682 + 0.730i)14-s + (−0.829 − 0.557i)15-s + (0.516 + 0.856i)16-s + (0.737 − 0.675i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.501 - 0.865i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.501 - 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6737019116 - 0.3883747374i\)
\(L(\frac12)\) \(\approx\) \(0.6737019116 - 0.3883747374i\)
\(L(1)\) \(\approx\) \(0.7539544651 - 0.1262470129i\)
\(L(1)\) \(\approx\) \(0.7539544651 - 0.1262470129i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.254 - 0.967i)T \)
3 \( 1 + (-0.511 + 0.859i)T \)
5 \( 1 + (-0.0552 + 0.998i)T \)
7 \( 1 + (-0.533 + 0.845i)T \)
11 \( 1 + (0.389 - 0.921i)T \)
13 \( 1 + (-0.992 - 0.123i)T \)
17 \( 1 + (0.737 - 0.675i)T \)
19 \( 1 + (-0.895 - 0.445i)T \)
23 \( 1 + (-0.977 - 0.212i)T \)
29 \( 1 + (0.811 - 0.584i)T \)
31 \( 1 + (-0.922 - 0.386i)T \)
37 \( 1 + (0.947 - 0.319i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (0.903 + 0.428i)T \)
47 \( 1 + (0.505 + 0.862i)T \)
53 \( 1 + (0.934 + 0.356i)T \)
59 \( 1 + (-0.687 + 0.726i)T \)
61 \( 1 + (-0.998 - 0.0455i)T \)
67 \( 1 + (0.126 - 0.991i)T \)
71 \( 1 + (0.710 - 0.703i)T \)
73 \( 1 + (0.934 - 0.356i)T \)
79 \( 1 + (-0.555 - 0.831i)T \)
83 \( 1 + (0.994 - 0.103i)T \)
89 \( 1 + (0.795 + 0.605i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−22.153121904978717000940248456667, −21.3253664586940544626054178111, −20.01124018230944099152572905940, −19.61054108057039252105611021414, −18.56463034005751670513842669312, −17.53285904837681243832906723656, −17.04484659719622595263601640705, −16.63772432283008033680448663699, −15.76855479933418347625734041877, −14.56523567788632060420359269071, −13.98115258061792926664037565335, −12.94297164858819047212338317400, −12.49628740137516498463276133825, −12.02891841811982657936741116488, −10.4113724589559142680630753004, −9.62652786061504468031353467265, −8.567896275945790975408169419076, −7.6900124032350957543416467768, −7.14894262002783891381165899828, −6.25288258196788718048315370803, −5.43700182241197046506737346971, −4.51702003012535968259222296157, −3.7951542447233566398542285689, −2.058915352183228710237267832314, −0.8382840646804087131515808786, 0.45540981703661275419245767305, 2.41321174221912081397486502050, 2.950203730396795851190154277587, 3.85392878295158130082442451671, 4.7760633666064135776169269324, 5.94081656336748678738991562229, 6.21681999682518495392854936450, 7.85062110621524593720485287441, 9.15136894703133863285121341666, 9.58802880828888223006438064747, 10.46861682876301578466477471857, 11.11951287690580768420082537326, 11.91014892623931730799110587747, 12.42810759983010259594387572310, 13.72159880439326168709436153221, 14.566541009241796226068698473431, 15.04230045932583948399639685385, 16.01893743796265388147522872849, 16.93131549077699085845522862671, 17.94919608056343049586231651670, 18.53901527387166753546553682928, 19.41015509967437658438147074748, 19.92628855911225368684963888159, 21.33853376090392945604819240850, 21.55170073484900821135045883265

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