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Properties

Label 1-207-207.83-r0-0-0
Degree $1$
Conductor $207$
Sign $0.998 - 0.0491i$
Analytic cond. $0.961303$
Root an. cond. $0.961303$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $0.998 - 0.0491i$
Analytic conductor: \(0.961303\)
Root analytic conductor: \(0.961303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 207,\ (0:\ ),\ 0.998 - 0.0491i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.035147897 - 0.05002049191i\)
\(L(\frac12)\) \(\approx\) \(2.035147897 - 0.05002049191i\)
\(L(1)\) \(\approx\) \(1.751932459 + 0.07997418948i\)
\(L(1)\) \(\approx\) \(1.751932459 + 0.07997418948i\)

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

−26.77295181425978616121826320645, −25.570602389298105610452858429347, −24.88077556949845124797153087454, −23.750258584005501492661104002559, −22.82204859947205549139570774619, −22.0268547824843270642402993754, −21.35597926425345919510944472984, −20.364817078094852370316530890181, −19.09743758802506977538403874995, −18.536891237929470308109550187729, −17.33945277264593613338143691069, −15.72301375052489911893067873198, −14.98030799170466774507740983429, −14.26614038484133270397350463194, −13.21525370051044878343628562960, −11.91032369877509643396560360726, −11.45025634218567518737331790841, −10.12034491147481036740361421945, −9.275487202535702532591389991746, −7.358950559582345586136959203130, −6.48404657289965017072425684183, −5.30623122495793940534893495211, −4.2224025272615418147582181692, −2.67771035922225177002229929525, −2.04218209222115420000528818439, 1.39851732226117411407707845844, 3.25183511766626891146838282400, 4.32709137624420511707770830931, 5.38476076083909453674233209702, 6.31639135568071966410344876742, 7.86419277211626016687978513270, 8.37840112863327420423710487632, 10.12058724117336725453114104708, 11.23335067721448988419053819758, 12.49247078877208434216495566896, 13.17489870129407726965643419010, 14.104402126700648910489574239, 15.041280184268830406863370106316, 16.41113483463742855657402453848, 16.80415261293851281504983331841, 17.76472062296002932961994428831, 19.50510091014071535576736416187, 20.42413509663715915139371576966, 21.13926230044048316947638943840, 22.03097580076383291270332059516, 23.27684464997504221310401635704, 23.8783866008388243344751160317, 24.72388780997398256823686187183, 25.498216342375281690751881299892, 26.69050508296348728971670077101

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