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Properties

Label 1-967-967.211-r1-0-0
Degree $1$
Conductor $967$
Sign $0.671 + 0.740i$
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  + (0.377 − 0.926i)2-s + (0.775 − 0.631i)3-s + (−0.715 − 0.699i)4-s + (0.877 + 0.480i)5-s + (−0.291 − 0.956i)6-s + (−0.538 + 0.842i)7-s + (−0.917 + 0.398i)8-s + (0.203 − 0.979i)9-s + (0.775 − 0.631i)10-s + (0.203 − 0.979i)11-s + (−0.995 − 0.0909i)12-s + (−0.746 + 0.665i)13-s + (0.576 + 0.816i)14-s + (0.983 − 0.181i)15-s + (0.0227 + 0.999i)16-s + (−0.334 − 0.942i)17-s + ⋯
L(s)  = 1  + (0.377 − 0.926i)2-s + (0.775 − 0.631i)3-s + (−0.715 − 0.699i)4-s + (0.877 + 0.480i)5-s + (−0.291 − 0.956i)6-s + (−0.538 + 0.842i)7-s + (−0.917 + 0.398i)8-s + (0.203 − 0.979i)9-s + (0.775 − 0.631i)10-s + (0.203 − 0.979i)11-s + (−0.995 − 0.0909i)12-s + (−0.746 + 0.665i)13-s + (0.576 + 0.816i)14-s + (0.983 − 0.181i)15-s + (0.0227 + 0.999i)16-s + (−0.334 − 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.037429777 + 0.4597792444i\)
\(L(\frac12)\) \(\approx\) \(1.037429777 + 0.4597792444i\)
\(L(1)\) \(\approx\) \(1.157210333 - 0.6452572812i\)
\(L(1)\) \(\approx\) \(1.157210333 - 0.6452572812i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.377 - 0.926i)T \)
3 \( 1 + (0.775 - 0.631i)T \)
5 \( 1 + (0.877 + 0.480i)T \)
7 \( 1 + (-0.538 + 0.842i)T \)
11 \( 1 + (0.203 - 0.979i)T \)
13 \( 1 + (-0.746 + 0.665i)T \)
17 \( 1 + (-0.334 - 0.942i)T \)
19 \( 1 + (0.158 + 0.987i)T \)
23 \( 1 + (-0.962 + 0.269i)T \)
29 \( 1 + (-0.203 - 0.979i)T \)
31 \( 1 + (0.803 + 0.595i)T \)
37 \( 1 + (-0.746 + 0.665i)T \)
41 \( 1 + (-0.682 + 0.730i)T \)
43 \( 1 + (0.648 + 0.761i)T \)
47 \( 1 + (-0.983 + 0.181i)T \)
53 \( 1 + (-0.247 + 0.968i)T \)
59 \( 1 + (-0.247 - 0.968i)T \)
61 \( 1 + (-0.974 - 0.225i)T \)
67 \( 1 + (0.917 + 0.398i)T \)
71 \( 1 + (-0.990 - 0.136i)T \)
73 \( 1 + (-0.247 - 0.968i)T \)
79 \( 1 + (0.247 + 0.968i)T \)
83 \( 1 + (0.291 + 0.956i)T \)
89 \( 1 + (-0.113 - 0.993i)T \)
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

−21.67667195563906091948629985334, −20.62112588953439971427657197870, −20.1206986617126750387605679041, −19.29020603111793433869686843719, −17.83513364604640292221774732710, −17.36303549108957587108795937960, −16.65102939985006051607363299498, −15.82751366567060013815970870611, −15.08546246349111933704589885852, −14.34785140737656565550369393516, −13.61289707109918872416834514010, −12.98902467289324729612409628931, −12.3095113414827368061690610693, −10.4995264885104752411946839768, −9.92603051340473010195655500804, −9.21185614171141519365911653778, −8.40445450059608578651957279128, −7.42574744684562519403424942129, −6.71085052661001418778811384343, −5.56887196343965593354610573885, −4.70374474263637505160207681127, −4.08186839567593680309865734817, −3.0145729000641503608675075803, −1.913687412482940582839886679000, −0.1677851963267164090561553994, 1.320433504283708336814028705633, 2.206720496750904946565206515527, 2.85792927084776203197402629137, 3.57607227962517659460373033021, 4.95521980652105063820158999937, 6.1213211705202058365982077748, 6.456183394679958909695691608571, 7.930399762718912199559352658672, 8.956687469763793153477844483911, 9.556498020864519929672491160255, 10.13191642706829737188217795539, 11.53561192455656357671989360615, 12.022751221700137333174110453, 12.936119438638823184697515233045, 13.83293505962217739755472217002, 14.05862085009678393502640235323, 14.931674810983623826591552119746, 15.92967646315618634716300058804, 17.24402563071486481363144610170, 18.19621940518694958918817643047, 18.72686387020291521891110039267, 19.217179141488569962499664075516, 20.02045700846650454935083062278, 21.04135747193921477647856153053, 21.4945908065957768279434743669

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