Properties

Label 1-967-967.449-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.138 + 0.990i$
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  + (−0.971 + 0.238i)2-s + (0.957 + 0.288i)3-s + (0.886 − 0.462i)4-s + (0.555 + 0.831i)5-s + (−0.998 − 0.0520i)6-s + (−0.985 + 0.168i)7-s + (−0.750 + 0.660i)8-s + (0.833 + 0.552i)9-s + (−0.737 − 0.675i)10-s + (0.353 − 0.935i)11-s + (0.982 − 0.187i)12-s + (0.986 + 0.161i)13-s + (0.917 − 0.398i)14-s + (0.291 + 0.956i)15-s + (0.571 − 0.820i)16-s + (0.560 − 0.828i)17-s + ⋯
L(s)  = 1  + (−0.971 + 0.238i)2-s + (0.957 + 0.288i)3-s + (0.886 − 0.462i)4-s + (0.555 + 0.831i)5-s + (−0.998 − 0.0520i)6-s + (−0.985 + 0.168i)7-s + (−0.750 + 0.660i)8-s + (0.833 + 0.552i)9-s + (−0.737 − 0.675i)10-s + (0.353 − 0.935i)11-s + (0.982 − 0.187i)12-s + (0.986 + 0.161i)13-s + (0.917 − 0.398i)14-s + (0.291 + 0.956i)15-s + (0.571 − 0.820i)16-s + (0.560 − 0.828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.138 + 0.990i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.327843606 + 1.526857136i\)
\(L(\frac12)\) \(\approx\) \(1.327843606 + 1.526857136i\)
\(L(1)\) \(\approx\) \(0.9943012923 + 0.3950799652i\)
\(L(1)\) \(\approx\) \(0.9943012923 + 0.3950799652i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.971 + 0.238i)T \)
3 \( 1 + (0.957 + 0.288i)T \)
5 \( 1 + (0.555 + 0.831i)T \)
7 \( 1 + (-0.985 + 0.168i)T \)
11 \( 1 + (0.353 - 0.935i)T \)
13 \( 1 + (0.986 + 0.161i)T \)
17 \( 1 + (0.560 - 0.828i)T \)
19 \( 1 + (-0.903 - 0.428i)T \)
23 \( 1 + (-0.993 - 0.116i)T \)
29 \( 1 + (-0.0876 + 0.996i)T \)
31 \( 1 + (0.999 + 0.0260i)T \)
37 \( 1 + (0.488 + 0.872i)T \)
41 \( 1 + (-0.854 + 0.519i)T \)
43 \( 1 + (0.00325 - 0.999i)T \)
47 \( 1 + (-0.929 - 0.368i)T \)
53 \( 1 + (0.983 - 0.181i)T \)
59 \( 1 + (0.471 + 0.881i)T \)
61 \( 1 + (0.0227 + 0.999i)T \)
67 \( 1 + (0.477 - 0.878i)T \)
71 \( 1 + (0.279 + 0.960i)T \)
73 \( 1 + (0.983 + 0.181i)T \)
79 \( 1 + (0.807 + 0.589i)T \)
83 \( 1 + (-0.272 + 0.962i)T \)
89 \( 1 + (0.522 + 0.852i)T \)
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

−20.995418519993265382060038339697, −20.42827372580202462365431189655, −19.67340776811971372833944469387, −19.19521585022565939076170621207, −18.261748115424953115822429456007, −17.48761626092743077291656458145, −16.72657197096245067926756952046, −15.90189714451785215463558389479, −15.21596191765796746985224403859, −14.11233831767958560670284917842, −13.03544420085700654046063849668, −12.70764902623473844813227768976, −11.86286627013896601327442089127, −10.30311256006516768600340719219, −9.89102165739131086876863265959, −9.1812747030894175642652746473, −8.34523985474562273323022448771, −7.78935827504050737834327557420, −6.497420247737533317095769456379, −6.10065526369606931394781864536, −4.18842296319979698896874421247, −3.51636264677013479321664853205, −2.2405161437657669338788763786, −1.61752123845594066272023911436, −0.56706522135187649128449187280, 1.007830359929614003941574109798, 2.23593858845633315986835680159, 3.001899594381241992089512737650, 3.71361509426078389906898864641, 5.479962425212470982109774577606, 6.494905834596877744315169528127, 6.86793230755863816742039778747, 8.164212674171285970460649010128, 8.766018284585844671948195604726, 9.595651064711543788642294272224, 10.1876199468797361188754129794, 10.92746086264032145504331857094, 11.94221509382260772978349346888, 13.37545411694256566758017726422, 13.86659109728761490566581362553, 14.78020624986503875165716633460, 15.52219803209274737987495677633, 16.27451891236954463508372289111, 16.85277488689066044402784543752, 18.30178489057998909449894211796, 18.51841595399853632528671112905, 19.337459742649005378147532538862, 19.91256206519715959777198011148, 20.93171888661442695469557989324, 21.55543196712252071475061380128

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