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Properties

Label 1-207-207.101-r1-0-0
Degree $1$
Conductor $207$
Sign $-0.998 + 0.0547i$
Analytic cond. $22.2452$
Root an. cond. $22.2452$
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  + (0.888 + 0.458i)2-s + (0.580 + 0.814i)4-s + (−0.235 + 0.971i)5-s + (−0.327 + 0.945i)7-s + (0.142 + 0.989i)8-s + (−0.654 + 0.755i)10-s + (−0.0475 + 0.998i)11-s + (−0.327 − 0.945i)13-s + (−0.723 + 0.690i)14-s + (−0.327 + 0.945i)16-s + (−0.415 − 0.909i)17-s + (0.415 − 0.909i)19-s + (−0.928 + 0.371i)20-s + (−0.5 + 0.866i)22-s + (−0.888 − 0.458i)25-s + (0.142 − 0.989i)26-s + ⋯
L(s)  = 1  + (0.888 + 0.458i)2-s + (0.580 + 0.814i)4-s + (−0.235 + 0.971i)5-s + (−0.327 + 0.945i)7-s + (0.142 + 0.989i)8-s + (−0.654 + 0.755i)10-s + (−0.0475 + 0.998i)11-s + (−0.327 − 0.945i)13-s + (−0.723 + 0.690i)14-s + (−0.327 + 0.945i)16-s + (−0.415 − 0.909i)17-s + (0.415 − 0.909i)19-s + (−0.928 + 0.371i)20-s + (−0.5 + 0.866i)22-s + (−0.888 − 0.458i)25-s + (0.142 − 0.989i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.998 + 0.0547i$
Analytic conductor: \(22.2452\)
Root analytic conductor: \(22.2452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 207,\ (1:\ ),\ -0.998 + 0.0547i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06088989228 + 2.221539754i\)
\(L(\frac12)\) \(\approx\) \(0.06088989228 + 2.221539754i\)
\(L(1)\) \(\approx\) \(1.124833286 + 1.011160987i\)
\(L(1)\) \(\approx\) \(1.124833286 + 1.011160987i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.888 + 0.458i)T \)
5 \( 1 + (-0.235 + 0.971i)T \)
7 \( 1 + (-0.327 + 0.945i)T \)
11 \( 1 + (-0.0475 + 0.998i)T \)
13 \( 1 + (-0.327 - 0.945i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (-0.580 + 0.814i)T \)
31 \( 1 + (0.928 + 0.371i)T \)
37 \( 1 + (-0.959 - 0.281i)T \)
41 \( 1 + (-0.235 + 0.971i)T \)
43 \( 1 + (0.928 - 0.371i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.327 + 0.945i)T \)
61 \( 1 + (-0.786 + 0.618i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (0.981 + 0.189i)T \)
83 \( 1 + (-0.235 - 0.971i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.723 + 0.690i)T \)
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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

−26.099275910625578034788951864109, −24.57658248289563871014272652718, −24.15075831271741858792757151668, −23.30449241422705134349674746172, −22.31401528202723505355469616278, −21.19556956781890908957654590610, −20.601055062011920645989885289081, −19.51602415835807187964226839058, −19.00453661106252988299831677753, −17.075950826884895811113790668314, −16.41664074852654765748464959097, −15.44213549066654205395268466604, −14.03879702411528673924976437815, −13.48212752149481484390123179512, −12.442772471102904490496456362194, −11.54200746258686251270424991706, −10.46721454861375518706608047848, −9.377036743427124353568367975232, −8.006185064461156976409010337237, −6.63655239250348228039735884016, −5.54336937311540600929350184728, −4.27579828239302629396294833435, −3.6073778690601269661268985442, −1.8127960745980040451495881427, −0.52923179038573479454548559370, 2.450173935634002089434157510786, 3.0989762931563182066048137671, 4.643829287136421297849781200185, 5.69324767346278550769956291577, 6.88470325795854723358131620069, 7.56315629892613295676803490307, 9.0399379499330956549682824418, 10.4337079367497165895109024514, 11.631466162409333356687020752, 12.40482923879787017544068103838, 13.47658135909714031800853390829, 14.63441549476830262596817426264, 15.365796190629209328511998707222, 15.916381084589472333030604723, 17.54861200479243137113104463289, 18.17876465563191806397922562295, 19.553805931206251277851572378087, 20.49100875361043585304930575693, 21.72845244980153924491686628147, 22.524371054145160804131446386298, 22.86822793027136824133347939376, 24.19520652428582266400869480314, 25.13849967792797638514404414900, 25.78308283636205113595677967814, 26.71995309787968083423907118883

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