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Properties

Label 1-1380-1380.707-r0-0-0
Degree $1$
Conductor $1380$
Sign $0.514 - 0.857i$
Analytic cond. $6.40869$
Root an. cond. $6.40869$
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  + (0.281 − 0.959i)7-s + (0.654 − 0.755i)11-s + (0.281 + 0.959i)13-s + (0.989 − 0.142i)17-s + (0.142 − 0.989i)19-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.909 − 0.415i)37-s + (−0.415 + 0.909i)41-s + (0.540 − 0.841i)43-s − i·47-s + (−0.841 − 0.540i)49-s + (−0.281 + 0.959i)53-s + (0.959 − 0.281i)59-s + (−0.841 + 0.540i)61-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)7-s + (0.654 − 0.755i)11-s + (0.281 + 0.959i)13-s + (0.989 − 0.142i)17-s + (0.142 − 0.989i)19-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.909 − 0.415i)37-s + (−0.415 + 0.909i)41-s + (0.540 − 0.841i)43-s − i·47-s + (−0.841 − 0.540i)49-s + (−0.281 + 0.959i)53-s + (0.959 − 0.281i)59-s + (−0.841 + 0.540i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.514 - 0.857i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ 0.514 - 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.492279408 - 0.8450404528i\)
\(L(\frac12)\) \(\approx\) \(1.492279408 - 0.8450404528i\)
\(L(1)\) \(\approx\) \(1.170854994 - 0.2429013638i\)
\(L(1)\) \(\approx\) \(1.170854994 - 0.2429013638i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (0.281 - 0.959i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (0.281 + 0.959i)T \)
17 \( 1 + (0.989 - 0.142i)T \)
19 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 + (0.909 - 0.415i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.281 + 0.959i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (0.755 - 0.654i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.989 + 0.142i)T \)
79 \( 1 + (0.959 - 0.281i)T \)
83 \( 1 + (-0.909 + 0.415i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (-0.909 - 0.415i)T \)
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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

−20.854476991306694003539688737524, −20.31754599954807062290619717316, −19.445262523383842910966282347530, −18.5048342670989751659796378743, −18.12486320678271161813126589673, −17.18718841765329832704507166185, −16.4293260596585063825203000404, −15.56725453522922511533002998563, −14.702175812232115346523404063371, −14.46291732750276559592697201855, −13.11496914862747231609453954109, −12.44168642684193162680036888503, −11.878177628184539364797101544643, −10.94569317875609826162434958885, −9.99957214401748147243812753183, −9.35384326748122253879010029588, −8.368483709263070344628244988514, −7.762029769666778194624719749498, −6.731676729290220630098186142294, −5.69449106253690892199699106093, −5.25862974137186159926898159574, −4.00676870541749847423518973603, −3.190396918608290468782941537652, −2.09092695720845246553319984399, −1.2093540697296856515039519403, 0.75796020192333398211056185903, 1.63654267288668217939640291578, 2.95618956255189249494542143042, 3.880781182663328868165854075754, 4.54306390677545548514588936490, 5.65830439804978362499195522942, 6.549270583304759097874194209524, 7.311862566230108824777005126095, 8.11921637887488549267543038780, 9.11303508192861220760960137478, 9.72570086303562819482157113051, 10.876235980231852268385106753997, 11.30025441818951675827542134295, 12.15688702917109715493074116595, 13.22087808241401102595194536554, 13.96069396138789142679541563801, 14.34934601179093011403882502967, 15.397679708126100214739593428644, 16.46910101110063722759728835537, 16.7330110137977097386872144964, 17.61655512525339303101137567154, 18.49638198938841140934935820092, 19.291427481477331209029217238, 19.871724956077861460986429387141, 20.76805567865457579133473178190

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