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Properties

Label 1-967-967.157-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.550 + 0.835i$
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  + (0.203 + 0.979i)2-s + (0.854 + 0.519i)3-s + (−0.917 + 0.398i)4-s + (0.682 − 0.730i)5-s + (−0.334 + 0.942i)6-s + (−0.0682 + 0.997i)7-s + (−0.576 − 0.816i)8-s + (0.460 + 0.887i)9-s + (0.854 + 0.519i)10-s + (0.460 + 0.887i)11-s + (−0.990 − 0.136i)12-s + (0.460 − 0.887i)13-s + (−0.990 + 0.136i)14-s + (0.962 − 0.269i)15-s + (0.682 − 0.730i)16-s + (0.962 + 0.269i)17-s + ⋯
L(s)  = 1  + (0.203 + 0.979i)2-s + (0.854 + 0.519i)3-s + (−0.917 + 0.398i)4-s + (0.682 − 0.730i)5-s + (−0.334 + 0.942i)6-s + (−0.0682 + 0.997i)7-s + (−0.576 − 0.816i)8-s + (0.460 + 0.887i)9-s + (0.854 + 0.519i)10-s + (0.460 + 0.887i)11-s + (−0.990 − 0.136i)12-s + (0.460 − 0.887i)13-s + (−0.990 + 0.136i)14-s + (0.962 − 0.269i)15-s + (0.682 − 0.730i)16-s + (0.962 + 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.550 + 0.835i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.550 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.177718167 + 2.185839138i\)
\(L(\frac12)\) \(\approx\) \(1.177718167 + 2.185839138i\)
\(L(1)\) \(\approx\) \(1.256437467 + 1.104440497i\)
\(L(1)\) \(\approx\) \(1.256437467 + 1.104440497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.203 + 0.979i)T \)
3 \( 1 + (0.854 + 0.519i)T \)
5 \( 1 + (0.682 - 0.730i)T \)
7 \( 1 + (-0.0682 + 0.997i)T \)
11 \( 1 + (0.460 + 0.887i)T \)
13 \( 1 + (0.460 - 0.887i)T \)
17 \( 1 + (0.962 + 0.269i)T \)
19 \( 1 + (0.854 + 0.519i)T \)
23 \( 1 + (-0.917 + 0.398i)T \)
29 \( 1 + (0.460 - 0.887i)T \)
31 \( 1 + (-0.576 - 0.816i)T \)
37 \( 1 + (0.460 - 0.887i)T \)
41 \( 1 + (-0.334 + 0.942i)T \)
43 \( 1 + (0.962 - 0.269i)T \)
47 \( 1 + (0.962 - 0.269i)T \)
53 \( 1 + (-0.917 + 0.398i)T \)
59 \( 1 + (-0.917 - 0.398i)T \)
61 \( 1 + (-0.334 + 0.942i)T \)
67 \( 1 + (-0.576 + 0.816i)T \)
71 \( 1 + (0.203 - 0.979i)T \)
73 \( 1 + (-0.917 - 0.398i)T \)
79 \( 1 + (-0.917 - 0.398i)T \)
83 \( 1 + (-0.334 + 0.942i)T \)
89 \( 1 + (-0.576 + 0.816i)T \)
97 \( 1 + T \)
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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

−21.45050053051637772462839730752, −20.55635994778349269405464074902, −20.01387493092045093985760504334, −19.10922451389962920351866938010, −18.61599541484128472679048711694, −17.920330993300505665254842487519, −16.99489732519861699866659298685, −15.887187890138441752958146323817, −14.3840933555952883171316885352, −14.09635183920763130211945418481, −13.784160514252815182322290809444, −12.811044164737954708501881004266, −11.83785023930670097496806970624, −10.98819117839827505205264576926, −10.17595110451676811139314174240, −9.41728487849262516811517413120, −8.70077977555767267513516452537, −7.5574510910368536101393808726, −6.66853363537566781634815034442, −5.79663567819101507475661596225, −4.39223883365787058806171743926, −3.393432707338409492282607305348, −3.001464503616713055897515587168, −1.71253077764963173394301661695, −1.052495416291011887390491787112, 1.44160652798519862561527952507, 2.64792312128189389897439322005, 3.74212363779750652069639328514, 4.60086532606147656419319960662, 5.63494389268344493636320007870, 5.96313101627815305192173879295, 7.60302396272901817225970269373, 8.06690322866321855992111837567, 9.04856201171947485411765994347, 9.56509608193236337788800822018, 10.18782076018390500866319385295, 12.01212545298034946630726379504, 12.65137774074515739821955197076, 13.45443441599293235810676252496, 14.25187625913806226389774025789, 14.91759386926162833045424768526, 15.68403872169814658802442778148, 16.24413514043449323283855298921, 17.160083653214278521268687689701, 17.99432491141171533785210813236, 18.66405579309462050692558535067, 19.79460585666134375248953242329, 20.641979643850024768980420878606, 21.268968358597781179181234180, 22.074489191105585344507256383242

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