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Properties

Label 1-967-967.336-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.867 + 0.497i$
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  + (0.787 − 0.615i)2-s + (−0.750 − 0.660i)3-s + (0.241 − 0.970i)4-s + (0.653 − 0.756i)5-s + (−0.998 − 0.0585i)6-s + (0.560 − 0.828i)7-s + (−0.407 − 0.913i)8-s + (0.126 + 0.991i)9-s + (0.0487 − 0.998i)10-s + (−0.544 − 0.838i)11-s + (−0.822 + 0.568i)12-s + (−0.544 + 0.838i)13-s + (−0.0682 − 0.997i)14-s + (−0.990 + 0.136i)15-s + (−0.883 − 0.468i)16-s + (0.951 − 0.307i)17-s + ⋯
L(s)  = 1  + (0.787 − 0.615i)2-s + (−0.750 − 0.660i)3-s + (0.241 − 0.970i)4-s + (0.653 − 0.756i)5-s + (−0.998 − 0.0585i)6-s + (0.560 − 0.828i)7-s + (−0.407 − 0.913i)8-s + (0.126 + 0.991i)9-s + (0.0487 − 0.998i)10-s + (−0.544 − 0.838i)11-s + (−0.822 + 0.568i)12-s + (−0.544 + 0.838i)13-s + (−0.0682 − 0.997i)14-s + (−0.990 + 0.136i)15-s + (−0.883 − 0.468i)16-s + (0.951 − 0.307i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.867 + 0.497i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (336, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.867 + 0.497i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4277219620 - 1.604796641i\)
\(L(\frac12)\) \(\approx\) \(-0.4277219620 - 1.604796641i\)
\(L(1)\) \(\approx\) \(0.7498959513 - 1.090312108i\)
\(L(1)\) \(\approx\) \(0.7498959513 - 1.090312108i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.787 - 0.615i)T \)
3 \( 1 + (-0.750 - 0.660i)T \)
5 \( 1 + (0.653 - 0.756i)T \)
7 \( 1 + (0.560 - 0.828i)T \)
11 \( 1 + (-0.544 - 0.838i)T \)
13 \( 1 + (-0.544 + 0.838i)T \)
17 \( 1 + (0.951 - 0.307i)T \)
19 \( 1 + (-0.477 + 0.878i)T \)
23 \( 1 + (-0.608 - 0.793i)T \)
29 \( 1 + (-0.995 + 0.0974i)T \)
31 \( 1 + (-0.0292 + 0.999i)T \)
37 \( 1 + (-0.696 - 0.717i)T \)
41 \( 1 + (-0.576 - 0.816i)T \)
43 \( 1 + (0.833 - 0.552i)T \)
47 \( 1 + (0.353 + 0.935i)T \)
53 \( 1 + (0.203 + 0.979i)T \)
59 \( 1 + (0.909 + 0.416i)T \)
61 \( 1 + (-0.576 - 0.816i)T \)
67 \( 1 + (-0.0292 - 0.999i)T \)
71 \( 1 + (0.787 + 0.615i)T \)
73 \( 1 + (0.203 - 0.979i)T \)
79 \( 1 + (0.892 - 0.451i)T \)
83 \( 1 + (0.874 + 0.485i)T \)
89 \( 1 + (-0.0292 - 0.999i)T \)
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

−22.298182910696574082328670814061, −21.59023633081947114327671241928, −21.08054187122129143891473368063, −20.29045894543575196470951602731, −18.75802096950479326934483853519, −17.76588677614586735429512919804, −17.586061096080604247558708838850, −16.687969655502064795536745372024, −15.56169838186824618089391527227, −14.97521936210842459286404181659, −14.77378665285344037994065145497, −13.44983048451838572445592033043, −12.64419454674329582500466340224, −11.83662717149997299387805613841, −11.10553205464124314480495040328, −10.13358231012552218922286216570, −9.41094374490094589627825130749, −8.10621127633802835249687999100, −7.26039348298652468740796886890, −6.29724425310519227404824159225, −5.39299406898988256503212047854, −5.19990761634553337085613233550, −3.957330628198049506304592216517, −2.90306947951915926377553009216, −2.022910500945599301214399246214, 0.57055320337102537479156259600, 1.56361860115266916429899600141, 2.24738831436210401074856038528, 3.76419405739202381089821545286, 4.74060215111829261474186665353, 5.414426912278214507728096843960, 6.09927453111285934056667024299, 7.11918632166740867621812193255, 8.08338510695981457606675799539, 9.295415022507020802746015660076, 10.50887361061038680577196672634, 10.70062052704381217426629474871, 12.00901977300296895528016248131, 12.334024602266670956087914822315, 13.26017889696836840265151192338, 14.081004004478105348161695815952, 14.282648708182151810593109268736, 16.00509720866410857893962201015, 16.612821168688112642594033617162, 17.25174012158258817331570498801, 18.35861615380128040382042282485, 18.91072848349122425433532005742, 19.82011080906567522707691579870, 20.82641810377507153809450813354, 21.17635342956133437948691327660

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