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Properties

Label 1-967-967.520-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: $\chi_{967} (520, \cdot )$
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

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

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