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Properties

Label 1-42e2-1764.1507-r1-0-0
Degree $1$
Conductor $1764$
Sign $-0.236 + 0.971i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (0.0747 − 0.997i)5-s + (−0.365 + 0.930i)11-s + (0.365 − 0.930i)13-s + (0.955 + 0.294i)17-s + (0.5 + 0.866i)19-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.955 + 0.294i)29-s − 31-s + (−0.733 + 0.680i)37-s + (0.826 + 0.563i)41-s + (−0.826 + 0.563i)43-s + (−0.623 − 0.781i)47-s + (−0.733 − 0.680i)53-s + (0.900 + 0.433i)55-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)5-s + (−0.365 + 0.930i)11-s + (0.365 − 0.930i)13-s + (0.955 + 0.294i)17-s + (0.5 + 0.866i)19-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.955 + 0.294i)29-s − 31-s + (−0.733 + 0.680i)37-s + (0.826 + 0.563i)41-s + (−0.826 + 0.563i)43-s + (−0.623 − 0.781i)47-s + (−0.733 − 0.680i)53-s + (0.900 + 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.236 + 0.971i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ -0.236 + 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6850856736 + 0.8717380813i\)
\(L(\frac12)\) \(\approx\) \(0.6850856736 + 0.8717380813i\)
\(L(1)\) \(\approx\) \(1.011658139 + 0.02979544360i\)
\(L(1)\) \(\approx\) \(1.011658139 + 0.02979544360i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.0747 - 0.997i)T \)
11 \( 1 + (-0.365 + 0.930i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
17 \( 1 + (0.955 + 0.294i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.733 + 0.680i)T \)
29 \( 1 + (0.955 + 0.294i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.733 + 0.680i)T \)
41 \( 1 + (0.826 + 0.563i)T \)
43 \( 1 + (-0.826 + 0.563i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (-0.733 - 0.680i)T \)
59 \( 1 + (0.900 + 0.433i)T \)
61 \( 1 + (-0.222 - 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (0.365 + 0.930i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.365 - 0.930i)T \)
89 \( 1 + (-0.988 - 0.149i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−19.573130200071436598632919782424, −19.06959232740108461982352549225, −18.44342818603572717991026762976, −17.81873709283647405583478866654, −16.82698902662406773408370404194, −16.11961216321650200058654239378, −15.47680004834727720006395730649, −14.42053366987139905666284112329, −14.06544199426801251497043303756, −13.31209202982964531269020238447, −12.29202441821891174420317064132, −11.37292590447330881387726882830, −10.944894011822280189523908209750, −10.12056690511130850396024479416, −9.22632439034597042496094839305, −8.47613502152925970514043644315, −7.43567196817930357148306030987, −6.8559213313230442523911144646, −5.99816764426958126663386986670, −5.21959801980836024003227131998, −4.09327143428885819345921058565, −3.16515836947080527317485023334, −2.60108403777743045735137966222, −1.37021108506272405960366520917, −0.208070719595620987616349670296, 1.09050685307550280538171720120, 1.68371144762339900796763979836, 3.03636957842520606453119725591, 3.80950588363873216722245611263, 5.00999052386760723067161861247, 5.31965816879624189324596051491, 6.32823054692202566105928528146, 7.512829892601350468845261384766, 8.0305441293911061594523338354, 8.844720738264011264348440292968, 9.84855735994859820901180585133, 10.20342764947884471246550870707, 11.37319705419435448394011458785, 12.23400864230827362114074946030, 12.777120137460049328870079699404, 13.36736504059817900736853053872, 14.39676882530958392368087827706, 15.129522131345639799350013203376, 15.93485115596166073107164330004, 16.53338348655565213642881497390, 17.370182811035219108244025655, 17.95145563547750127647061194010, 18.748487621678979074401748231151, 19.78089369143994827203205213253, 20.23368885351853773092806752625

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