↑

Properties

Label 1-967-967.179-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.0366 + 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.986 − 0.161i)2-s + (0.425 − 0.905i)3-s + (0.947 + 0.319i)4-s + (0.581 + 0.813i)5-s + (−0.566 + 0.824i)6-s + (0.701 + 0.712i)7-s + (−0.883 − 0.468i)8-s + (−0.638 − 0.769i)9-s + (−0.442 − 0.896i)10-s + (0.241 + 0.970i)11-s + (0.692 − 0.721i)12-s + (−0.961 + 0.276i)13-s + (−0.576 − 0.816i)14-s + (0.983 − 0.181i)15-s + (0.795 + 0.605i)16-s + (−0.107 + 0.994i)17-s + ⋯
L(s)  = 1  + (−0.986 − 0.161i)2-s + (0.425 − 0.905i)3-s + (0.947 + 0.319i)4-s + (0.581 + 0.813i)5-s + (−0.566 + 0.824i)6-s + (0.701 + 0.712i)7-s + (−0.883 − 0.468i)8-s + (−0.638 − 0.769i)9-s + (−0.442 − 0.896i)10-s + (0.241 + 0.970i)11-s + (0.692 − 0.721i)12-s + (−0.961 + 0.276i)13-s + (−0.576 − 0.816i)14-s + (0.983 − 0.181i)15-s + (0.795 + 0.605i)16-s + (−0.107 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0366 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0366 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.0366 + 0.999i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.0366 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5673300155 + 0.5885192417i\)
\(L(\frac12)\) \(\approx\) \(0.5673300155 + 0.5885192417i\)
\(L(1)\) \(\approx\) \(0.7808924509 + 0.06885533600i\)
\(L(1)\) \(\approx\) \(0.7808924509 + 0.06885533600i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.986 - 0.161i)T \)
3 \( 1 + (0.425 - 0.905i)T \)
5 \( 1 + (0.581 + 0.813i)T \)
7 \( 1 + (0.701 + 0.712i)T \)
11 \( 1 + (0.241 + 0.970i)T \)
13 \( 1 + (-0.961 + 0.276i)T \)
17 \( 1 + (-0.107 + 0.994i)T \)
19 \( 1 + (-0.927 + 0.374i)T \)
23 \( 1 + (-0.984 - 0.174i)T \)
29 \( 1 + (-0.608 - 0.793i)T \)
31 \( 1 + (-0.465 - 0.884i)T \)
37 \( 1 + (0.483 + 0.875i)T \)
41 \( 1 + (0.682 - 0.730i)T \)
43 \( 1 + (0.254 + 0.967i)T \)
47 \( 1 + (-0.0422 + 0.999i)T \)
53 \( 1 + (-0.247 + 0.968i)T \)
59 \( 1 + (-0.889 + 0.457i)T \)
61 \( 1 + (-0.974 - 0.225i)T \)
67 \( 1 + (0.999 - 0.0390i)T \)
71 \( 1 + (0.353 - 0.935i)T \)
73 \( 1 + (-0.247 - 0.968i)T \)
79 \( 1 + (-0.911 - 0.410i)T \)
83 \( 1 + (0.152 - 0.988i)T \)
89 \( 1 + (-0.533 - 0.845i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.50669125149848746768657417886, −20.516112938196375465253234980618, −20.03556936258931140560980641643, −19.4840873799564153556146763836, −18.20622560820780923855054525232, −17.455682775556939963431446482041, −16.70975669906785365366765330213, −16.35975431024418154953051293438, −15.4177280857491492099478863016, −14.34576948341728887906549673720, −14.01434404035271983297833554841, −12.72184062084601399315979836519, −11.51306625357022111413807783013, −10.84785688754308496940060377049, −10.0479595088366263912648759117, −9.33141479612008227609706780229, −8.631819902777221100819110121412, −7.96664901674456330065567673814, −6.97554030766417845796796622909, −5.6475428250612234223792761145, −5.0292597349267930510552672779, −3.94389415109873394120707315859, −2.66572205261691285622720974500, −1.74827981053160497232937668648, −0.41078189771846528588792989008, 1.73209242325006347007664312388, 2.01912479504055884902578147107, 2.79468727966165255743692454918, 4.21774903482119067220351117500, 6.01352072582362418231128683313, 6.32147610151907112955129794951, 7.597879862549062760087605204013, 7.818015165129935626386779999947, 9.039433279833685621955422676842, 9.62745350269793614095689469430, 10.584457960541144371548544149475, 11.51801819404828420990435257518, 12.26352054501542777434143206564, 12.9083201599976211387646677448, 14.27779625502689334973755781966, 14.85105927518965421543706952914, 15.32522747119038339760657525429, 17.07936553384692470485683628365, 17.32736902548086831822555204995, 18.153676089954569059573101017212, 18.72791223892787792838013920473, 19.35490383732304853171586056808, 20.17053392379486256713224586887, 21.02398006601234558346448589913, 21.7433370491279344804571743562

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