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Properties

Label 1-967-967.96-r1-0-0
Degree $1$
Conductor $967$
Sign $0.174 - 0.984i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (0.0227 − 0.999i)2-s + (0.917 − 0.398i)3-s + (−0.998 − 0.0455i)4-s + (0.419 + 0.907i)5-s + (−0.377 − 0.926i)6-s + (0.648 + 0.761i)7-s + (−0.0682 + 0.997i)8-s + (0.682 − 0.730i)9-s + (0.917 − 0.398i)10-s + (0.682 − 0.730i)11-s + (−0.934 + 0.356i)12-s + (0.974 − 0.225i)13-s + (0.775 − 0.631i)14-s + (0.746 + 0.665i)15-s + (0.995 + 0.0909i)16-s + (0.203 + 0.979i)17-s + ⋯
L(s)  = 1  + (0.0227 − 0.999i)2-s + (0.917 − 0.398i)3-s + (−0.998 − 0.0455i)4-s + (0.419 + 0.907i)5-s + (−0.377 − 0.926i)6-s + (0.648 + 0.761i)7-s + (−0.0682 + 0.997i)8-s + (0.682 − 0.730i)9-s + (0.917 − 0.398i)10-s + (0.682 − 0.730i)11-s + (−0.934 + 0.356i)12-s + (0.974 − 0.225i)13-s + (0.775 − 0.631i)14-s + (0.746 + 0.665i)15-s + (0.995 + 0.0909i)16-s + (0.203 + 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.174 - 0.984i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (96, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.174 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.923609018 - 2.450950423i\)
\(L(\frac12)\) \(\approx\) \(2.923609018 - 2.450950423i\)
\(L(1)\) \(\approx\) \(1.535074498 - 0.8007758531i\)
\(L(1)\) \(\approx\) \(1.535074498 - 0.8007758531i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.0227 - 0.999i)T \)
3 \( 1 + (0.917 - 0.398i)T \)
5 \( 1 + (0.419 + 0.907i)T \)
7 \( 1 + (0.648 + 0.761i)T \)
11 \( 1 + (0.682 - 0.730i)T \)
13 \( 1 + (0.974 - 0.225i)T \)
17 \( 1 + (0.203 + 0.979i)T \)
19 \( 1 + (-0.803 - 0.595i)T \)
23 \( 1 + (-0.460 - 0.887i)T \)
29 \( 1 + (-0.682 - 0.730i)T \)
31 \( 1 + (-0.829 - 0.557i)T \)
37 \( 1 + (0.974 - 0.225i)T \)
41 \( 1 + (0.990 + 0.136i)T \)
43 \( 1 + (0.949 - 0.313i)T \)
47 \( 1 + (-0.746 - 0.665i)T \)
53 \( 1 + (0.538 - 0.842i)T \)
59 \( 1 + (0.538 + 0.842i)T \)
61 \( 1 + (0.613 - 0.789i)T \)
67 \( 1 + (0.0682 + 0.997i)T \)
71 \( 1 + (0.854 - 0.519i)T \)
73 \( 1 + (0.538 + 0.842i)T \)
79 \( 1 + (-0.538 - 0.842i)T \)
83 \( 1 + (0.377 + 0.926i)T \)
89 \( 1 + (-0.898 - 0.439i)T \)
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

−21.56174700832520470570406777938, −20.955607752115407481026370485567, −20.25601108583267505895940460535, −19.517758329430826916971548441004, −18.32482383980478888003965023339, −17.69761116409545286348069744453, −16.70108710234133916042449702927, −16.30846342095436589513448414529, −15.414188422682627119986036030364, −14.350340681617480736662258275135, −14.146150444614439167340317124470, −13.218009854183513198811619612547, −12.55947272695834680806706135219, −11.13717261333910991827294719916, −9.961941151349609933065560317573, −9.33125141508335691217603954029, −8.69054859534533359744454605121, −7.82873225810163151122836052872, −7.19162514647632751245678555588, −5.98808775390014538387251180023, −4.95928660946222104209391860840, −4.25717324636848150661325870970, −3.64505118225867573560036948049, −1.841645236542499283352960491193, −1.03727613566539666465462572321, 0.83665173739689808930062155186, 2.008717771513796733662106808672, 2.40955545615851561985068140303, 3.55780950521875633327563788696, 4.12098475099046932036601190892, 5.75763888857420933490514890209, 6.33979086656860691458027908040, 7.78939414950670373351864523735, 8.55416910673093371497203696420, 9.10110654672108945131249170040, 10.101018477937338630925805675, 11.04533604420015240457435200391, 11.5446624591377474834878504200, 12.7840122836956752261142742318, 13.24429961788884719131013689015, 14.36957475625367982128624337621, 14.54643447562141126802302735972, 15.43544320697922633913380815720, 16.96808517912672970625997243204, 17.969257963775775114587721140125, 18.39397573790631431330237772740, 19.09892859917411262766721073106, 19.640885210583906107751887648882, 20.73149636990194605471573863762, 21.30300378156397386251565078275

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