Properties

Label 1-967-967.78-r0-0-0
Degree $1$
Conductor $967$
Sign $0.759 - 0.651i$
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  + (−0.883 + 0.468i)2-s + (−0.967 − 0.250i)3-s + (0.560 − 0.828i)4-s + (−0.957 − 0.288i)5-s + (0.972 − 0.232i)6-s + (−0.724 − 0.689i)7-s + (−0.107 + 0.994i)8-s + (0.874 + 0.485i)9-s + (0.981 − 0.193i)10-s + (−0.668 + 0.744i)11-s + (−0.750 + 0.660i)12-s + (−0.668 − 0.744i)13-s + (0.962 + 0.269i)14-s + (0.854 + 0.519i)15-s + (−0.371 − 0.928i)16-s + (0.316 + 0.948i)17-s + ⋯
L(s)  = 1  + (−0.883 + 0.468i)2-s + (−0.967 − 0.250i)3-s + (0.560 − 0.828i)4-s + (−0.957 − 0.288i)5-s + (0.972 − 0.232i)6-s + (−0.724 − 0.689i)7-s + (−0.107 + 0.994i)8-s + (0.874 + 0.485i)9-s + (0.981 − 0.193i)10-s + (−0.668 + 0.744i)11-s + (−0.750 + 0.660i)12-s + (−0.668 − 0.744i)13-s + (0.962 + 0.269i)14-s + (0.854 + 0.519i)15-s + (−0.371 − 0.928i)16-s + (0.316 + 0.948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.759 - 0.651i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.759 - 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2320620486 - 0.08589012472i\)
\(L(\frac12)\) \(\approx\) \(0.2320620486 - 0.08589012472i\)
\(L(1)\) \(\approx\) \(0.3507909954 + 0.001872042308i\)
\(L(1)\) \(\approx\) \(0.3507909954 + 0.001872042308i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.883 + 0.468i)T \)
3 \( 1 + (-0.967 - 0.250i)T \)
5 \( 1 + (-0.957 - 0.288i)T \)
7 \( 1 + (-0.724 - 0.689i)T \)
11 \( 1 + (-0.668 + 0.744i)T \)
13 \( 1 + (-0.668 - 0.744i)T \)
17 \( 1 + (0.316 + 0.948i)T \)
19 \( 1 + (-0.407 - 0.913i)T \)
23 \( 1 + (-0.864 + 0.502i)T \)
29 \( 1 + (0.924 + 0.380i)T \)
31 \( 1 + (0.993 - 0.116i)T \)
37 \( 1 + (-0.998 + 0.0585i)T \)
41 \( 1 + (-0.775 + 0.631i)T \)
43 \( 1 + (-0.696 + 0.717i)T \)
47 \( 1 + (0.126 + 0.991i)T \)
53 \( 1 + (0.682 + 0.730i)T \)
59 \( 1 + (-0.145 - 0.989i)T \)
61 \( 1 + (-0.775 + 0.631i)T \)
67 \( 1 + (0.993 + 0.116i)T \)
71 \( 1 + (-0.883 - 0.468i)T \)
73 \( 1 + (0.682 - 0.730i)T \)
79 \( 1 + (-0.297 + 0.954i)T \)
83 \( 1 + (-0.442 - 0.896i)T \)
89 \( 1 + (0.993 + 0.116i)T \)
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

−21.72559676789046991495647423372, −21.16611434181670976676403682082, −20.15853462309283640616864722841, −19.08668873344866468942424282978, −18.795378129802468968405712298301, −18.17189299097515412260462129992, −17.02271768792203536500243882393, −16.29930738313539555943599946225, −15.90965712408094552441899177485, −15.1667536719723865234654780393, −13.735577153632596801091775392133, −12.404859700806393755110730940284, −12.022052099467676251049471913670, −11.532959586159133775236813812493, −10.31405918296691503383920999835, −10.093531431695205872364470766, −8.85434989042224232081347555945, −8.09241453897127255882873746669, −7.00773454526434848947588673245, −6.4329972477982390419653680961, −5.25454437544825446499684167489, −4.07294909629449139631246080272, −3.21160920044700176716748270974, −2.20403295711618435487769035048, −0.53978696007433512670963882680, 0.342001406361305931884509951197, 1.46444624685712724764084767399, 2.92378286054392781593827540714, 4.40181064413239544563618078092, 5.10590805126170643087017670089, 6.229134934827902700520921598493, 6.99364157481648336456064769165, 7.669503650257939198485281194053, 8.32198451236170271878449864002, 9.7554003343957544072769786401, 10.3067857239410979972877303930, 10.95872879685315930672702935292, 12.05177743308442133199393315343, 12.60187301188121114038347331414, 13.57706260436842922683460594861, 15.02546931599706613381594297285, 15.60519219351893155457179452538, 16.18341514026014869367642101885, 17.12128830830643362708223951531, 17.484598000445541519637310106184, 18.39329823960770775124893847960, 19.42961642181546037466619246480, 19.65923536473585082589334256519, 20.56942101536055624090877567646, 21.7885305827427744005500649854

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