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Properties

Label 1-967-967.3-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.623 - 0.781i$
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  + (0.0487 − 0.998i)2-s + (0.608 + 0.793i)3-s + (−0.995 − 0.0974i)4-s + (0.0292 + 0.999i)5-s + (0.822 − 0.568i)6-s + (−0.924 + 0.380i)7-s + (−0.145 + 0.989i)8-s + (−0.260 + 0.965i)9-s + (0.999 + 0.0195i)10-s + (0.653 − 0.756i)11-s + (−0.527 − 0.849i)12-s + (−0.653 − 0.756i)13-s + (0.334 + 0.942i)14-s + (−0.775 + 0.631i)15-s + (0.981 + 0.193i)16-s + (0.425 + 0.905i)17-s + ⋯
L(s)  = 1  + (0.0487 − 0.998i)2-s + (0.608 + 0.793i)3-s + (−0.995 − 0.0974i)4-s + (0.0292 + 0.999i)5-s + (0.822 − 0.568i)6-s + (−0.924 + 0.380i)7-s + (−0.145 + 0.989i)8-s + (−0.260 + 0.965i)9-s + (0.999 + 0.0195i)10-s + (0.653 − 0.756i)11-s + (−0.527 − 0.849i)12-s + (−0.653 − 0.756i)13-s + (0.334 + 0.942i)14-s + (−0.775 + 0.631i)15-s + (0.981 + 0.193i)16-s + (0.425 + 0.905i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.623 - 0.781i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.623 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09215004247 - 0.1913162437i\)
\(L(\frac12)\) \(\approx\) \(0.09215004247 - 0.1913162437i\)
\(L(1)\) \(\approx\) \(0.8861139379 + 0.02531825388i\)
\(L(1)\) \(\approx\) \(0.8861139379 + 0.02531825388i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.0487 - 0.998i)T \)
3 \( 1 + (0.608 + 0.793i)T \)
5 \( 1 + (0.0292 + 0.999i)T \)
7 \( 1 + (-0.924 + 0.380i)T \)
11 \( 1 + (0.653 - 0.756i)T \)
13 \( 1 + (-0.653 - 0.756i)T \)
17 \( 1 + (0.425 + 0.905i)T \)
19 \( 1 + (-0.909 + 0.416i)T \)
23 \( 1 + (0.544 + 0.838i)T \)
29 \( 1 + (-0.999 + 0.0390i)T \)
31 \( 1 + (-0.297 - 0.954i)T \)
37 \( 1 + (-0.592 + 0.805i)T \)
41 \( 1 + (0.0682 - 0.997i)T \)
43 \( 1 + (-0.972 + 0.232i)T \)
47 \( 1 + (0.442 + 0.896i)T \)
53 \( 1 + (0.854 + 0.519i)T \)
59 \( 1 + (-0.696 - 0.717i)T \)
61 \( 1 + (-0.0682 + 0.997i)T \)
67 \( 1 + (0.297 - 0.954i)T \)
71 \( 1 + (0.0487 + 0.998i)T \)
73 \( 1 + (0.854 - 0.519i)T \)
79 \( 1 + (-0.126 + 0.991i)T \)
83 \( 1 + (0.494 - 0.869i)T \)
89 \( 1 + (0.297 - 0.954i)T \)
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

−21.98224123101972588032478831940, −20.98744802773820635800540501243, −20.01020843267315461266193241403, −19.48219815804734875100187122438, −18.68143592321119865870477123793, −17.76973960017909147894287569410, −16.77632566707849504276560292693, −16.65608550002067448692634696314, −15.436926721556949448444943046691, −14.6441467380653304714109934668, −13.91341679854313361202641507809, −13.10639223522886921054840203481, −12.57015080240153367904498565269, −11.902972694426583831319294484049, −10.05545011131478872783787778510, −9.14581266042228036560333578656, −8.977687539503759357548963298577, −7.770338256566866685396089470153, −6.933195568163546314109732932451, −6.58003630865626823871001967181, −5.28558162735449294986374413884, −4.3721104598375889256668356816, −3.51576299676737355357663878488, −2.10917501510370872414153558958, −0.87193355716070127478552593789, 0.04835413411733694028898872385, 1.83825917732761014889945351239, 2.79271637659645995422198040296, 3.462630337551007503391602488420, 3.95973986277859090799769216306, 5.42015120583590306325518018318, 6.11582941600923959486544501175, 7.55726941971334264613444812248, 8.520764321343163403015966107344, 9.359732318507293913149149280506, 10.028270675195607314321398265473, 10.6556838356025652176008150247, 11.400222196511711244719592834690, 12.46207827813822229131600596109, 13.30845054295297654919833237229, 14.06969842064680950805589528267, 15.01161895408119710768017652792, 15.24286633994884612071563511107, 16.740833473301056209981442470202, 17.259938824951551339130649417525, 18.66520615911624691264713594535, 19.11903734697955568859762555964, 19.578746856553523898729808633792, 20.46162617129429280576341585779, 21.51863691495528106925644526378

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