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Properties

Label 1-967-967.106-r0-0-0
Degree $1$
Conductor $967$
Sign $0.471 - 0.881i$
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  + (−0.587 + 0.809i)2-s + (0.560 − 0.828i)3-s + (−0.310 − 0.950i)4-s + (0.886 − 0.462i)5-s + (0.341 + 0.940i)6-s + (0.672 − 0.739i)7-s + (0.951 + 0.307i)8-s + (−0.371 − 0.928i)9-s + (−0.145 + 0.989i)10-s + (0.987 − 0.155i)11-s + (−0.961 − 0.276i)12-s + (−0.628 + 0.777i)13-s + (0.203 + 0.979i)14-s + (0.113 − 0.993i)15-s + (−0.807 + 0.589i)16-s + (0.592 − 0.805i)17-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (0.560 − 0.828i)3-s + (−0.310 − 0.950i)4-s + (0.886 − 0.462i)5-s + (0.341 + 0.940i)6-s + (0.672 − 0.739i)7-s + (0.951 + 0.307i)8-s + (−0.371 − 0.928i)9-s + (−0.145 + 0.989i)10-s + (0.987 − 0.155i)11-s + (−0.961 − 0.276i)12-s + (−0.628 + 0.777i)13-s + (0.203 + 0.979i)14-s + (0.113 − 0.993i)15-s + (−0.807 + 0.589i)16-s + (0.592 − 0.805i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.498752916 - 0.8984628623i\)
\(L(\frac12)\) \(\approx\) \(1.498752916 - 0.8984628623i\)
\(L(1)\) \(\approx\) \(1.191399813 - 0.2536815736i\)
\(L(1)\) \(\approx\) \(1.191399813 - 0.2536815736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.587 + 0.809i)T \)
3 \( 1 + (0.560 - 0.828i)T \)
5 \( 1 + (0.886 - 0.462i)T \)
7 \( 1 + (0.672 - 0.739i)T \)
11 \( 1 + (0.987 - 0.155i)T \)
13 \( 1 + (-0.628 + 0.777i)T \)
17 \( 1 + (0.592 - 0.805i)T \)
19 \( 1 + (-0.566 - 0.824i)T \)
23 \( 1 + (0.924 - 0.380i)T \)
29 \( 1 + (-0.957 + 0.288i)T \)
31 \( 1 + (0.818 + 0.574i)T \)
37 \( 1 + (0.216 + 0.976i)T \)
41 \( 1 + (0.962 - 0.269i)T \)
43 \( 1 + (-0.759 + 0.651i)T \)
47 \( 1 + (0.847 - 0.530i)T \)
53 \( 1 + (-0.419 + 0.907i)T \)
59 \( 1 + (-0.971 - 0.238i)T \)
61 \( 1 + (-0.247 + 0.968i)T \)
67 \( 1 + (0.0876 + 0.996i)T \)
71 \( 1 + (-0.407 + 0.913i)T \)
73 \( 1 + (-0.419 - 0.907i)T \)
79 \( 1 + (0.771 + 0.636i)T \)
83 \( 1 + (0.840 - 0.541i)T \)
89 \( 1 + (-0.906 - 0.422i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−21.6842179488578832992287871649, −21.040419664958612116932822896929, −20.54460529909304035693487108801, −19.41143745544553950019641315306, −18.99403968918755208265289744753, −17.978709128039002226243287018850, −17.09975927263671496610158806523, −16.852319575190388166404356274584, −15.32038151203353518687825984816, −14.73152921263371848037529975657, −14.07162011832896340905231894018, −12.98630247721684136700951124674, −12.180481235931626208644732251092, −11.10538672507098782648706624775, −10.55544676218704453509596699570, −9.61467631993224010930421631339, −9.24558116986924495613332832854, −8.25169264995793867271001875083, −7.545303934344936560926088500384, −6.047260946829229607910931272849, −5.120696445946801359544496407760, −4.04635187432389741153125982704, −3.146686923470713574535886020216, −2.25550290888378309497260979794, −1.55788515892380921651225408813, 0.95067632492274941159211657073, 1.50858743488688829582583642266, 2.62505597426195261170242865960, 4.308661791460842665517364219222, 5.08724668085497535104482523553, 6.27413968862983282543301809688, 6.92078091821387538566013878679, 7.57130983301951768197520536959, 8.69181264041829793745257270267, 9.12546128136619853368118502424, 9.918115435289370226948307028609, 11.07531960134897037770423073277, 12.04082349679985894682311823624, 13.22395545538951358555555197288, 13.84580321980830009995984412334, 14.41075753607832119128133524678, 15.00011581086556310315915330274, 16.471600352432811183841230511379, 17.08543073948503009016491590135, 17.471148194920129044127691903678, 18.39946394162676474872041007259, 19.12314147183395284124089217108, 19.91712717829573733847168921346, 20.57167499782103765631861581554, 21.49037934229180585326180974307

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