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Properties

Label 1-967-967.11-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.716 + 0.697i$
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  + (0.316 − 0.948i)2-s + (0.653 − 0.756i)3-s + (−0.799 − 0.600i)4-s + (0.874 − 0.485i)5-s + (−0.511 − 0.859i)6-s + (−0.844 − 0.536i)7-s + (−0.822 + 0.568i)8-s + (−0.145 − 0.989i)9-s + (−0.184 − 0.982i)10-s + (−0.297 + 0.954i)11-s + (−0.977 + 0.212i)12-s + (−0.297 − 0.954i)13-s + (−0.775 + 0.631i)14-s + (0.203 − 0.979i)15-s + (0.279 + 0.960i)16-s + (0.241 − 0.970i)17-s + ⋯
L(s)  = 1  + (0.316 − 0.948i)2-s + (0.653 − 0.756i)3-s + (−0.799 − 0.600i)4-s + (0.874 − 0.485i)5-s + (−0.511 − 0.859i)6-s + (−0.844 − 0.536i)7-s + (−0.822 + 0.568i)8-s + (−0.145 − 0.989i)9-s + (−0.184 − 0.982i)10-s + (−0.297 + 0.954i)11-s + (−0.977 + 0.212i)12-s + (−0.297 − 0.954i)13-s + (−0.775 + 0.631i)14-s + (0.203 − 0.979i)15-s + (0.279 + 0.960i)16-s + (0.241 − 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.716 + 0.697i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.716 + 0.697i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6155014965 - 1.513766888i\)
\(L(\frac12)\) \(\approx\) \(-0.6155014965 - 1.513766888i\)
\(L(1)\) \(\approx\) \(0.6365858402 - 1.149418194i\)
\(L(1)\) \(\approx\) \(0.6365858402 - 1.149418194i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.316 - 0.948i)T \)
3 \( 1 + (0.653 - 0.756i)T \)
5 \( 1 + (0.874 - 0.485i)T \)
7 \( 1 + (-0.844 - 0.536i)T \)
11 \( 1 + (-0.297 + 0.954i)T \)
13 \( 1 + (-0.297 - 0.954i)T \)
17 \( 1 + (0.241 - 0.970i)T \)
19 \( 1 + (0.592 + 0.805i)T \)
23 \( 1 + (-0.0292 - 0.999i)T \)
29 \( 1 + (-0.932 - 0.362i)T \)
31 \( 1 + (0.494 - 0.869i)T \)
37 \( 1 + (-0.864 + 0.502i)T \)
41 \( 1 + (-0.990 - 0.136i)T \)
43 \( 1 + (-0.608 + 0.793i)T \)
47 \( 1 + (0.909 + 0.416i)T \)
53 \( 1 + (0.460 + 0.887i)T \)
59 \( 1 + (-0.967 - 0.250i)T \)
61 \( 1 + (-0.990 - 0.136i)T \)
67 \( 1 + (0.494 + 0.869i)T \)
71 \( 1 + (0.316 + 0.948i)T \)
73 \( 1 + (0.460 - 0.887i)T \)
79 \( 1 + (-0.407 - 0.913i)T \)
83 \( 1 + (0.833 + 0.552i)T \)
89 \( 1 + (0.494 + 0.869i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−22.009119386432384098870798403365, −21.603288973189776184752925778793, −21.24255814082694812671209009856, −19.778903172311771785151909049807, −18.96131422696659689875006737354, −18.37554069665079616601495287325, −17.1365347153696355675826485393, −16.65462859991852710651100278477, −15.71502609571565009807649112014, −15.2437191526083970525171908199, −14.29174631604429172230710756293, −13.68019545250047235798769407467, −13.18648343354817834590524909578, −11.96478774491436129177412995803, −10.72429649338227789703871489019, −9.81725461354772419329048630142, −9.12099109644461799132753321051, −8.62115584759079726524236714976, −7.37966412791203052477847271950, −6.54471934978788831930884655922, −5.62341135254506730494687479472, −5.06175699069240614431819852982, −3.58747450646249244619228896818, −3.22103607207211770928520091071, −2.03331044626434985766164379636, 0.55375651117786464899395637314, 1.584142107017530594402435108786, 2.544299351669246311465278874335, 3.19301748771816800815149107981, 4.36601237376853114304233455283, 5.40504260493552734634192126345, 6.27119808004809871007527275673, 7.35119302269354338019598118860, 8.29823811697155445926350090601, 9.49736132419481276654233363845, 9.75019177627988391820456234786, 10.57243574693858699969120813250, 12.15253444846172163777473887677, 12.41951802953384164153931035280, 13.35374616501565538697209070781, 13.65243992875791627309357721838, 14.58333625913644468697347486011, 15.442002894025817303205609697622, 16.80040268463339909878688866551, 17.56995977384747902266534300946, 18.40271494202728015617127942474, 18.87919342528331435013475198756, 20.08612288070976312697296525722, 20.45687102282381063040121558317, 20.736257053329664936278268718127

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