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Properties

Label 1-1380-1380.527-r0-0-0
Degree $1$
Conductor $1380$
Sign $0.996 - 0.0830i$
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  + (0.989 − 0.142i)7-s + (−0.415 + 0.909i)11-s + (0.989 + 0.142i)13-s + (0.755 − 0.654i)17-s + (0.654 − 0.755i)19-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.540 − 0.841i)37-s + (−0.841 − 0.540i)41-s + (0.281 + 0.959i)43-s − i·47-s + (0.959 − 0.281i)49-s + (−0.989 + 0.142i)53-s + (0.142 − 0.989i)59-s + (0.959 + 0.281i)61-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)7-s + (−0.415 + 0.909i)11-s + (0.989 + 0.142i)13-s + (0.755 − 0.654i)17-s + (0.654 − 0.755i)19-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.540 − 0.841i)37-s + (−0.841 − 0.540i)41-s + (0.281 + 0.959i)43-s − i·47-s + (0.959 − 0.281i)49-s + (−0.989 + 0.142i)53-s + (0.142 − 0.989i)59-s + (0.959 + 0.281i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.996 - 0.0830i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ 0.996 - 0.0830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.907960960 - 0.07938632160i\)
\(L(\frac12)\) \(\approx\) \(1.907960960 - 0.07938632160i\)
\(L(1)\) \(\approx\) \(1.280604505 + 0.01336219777i\)
\(L(1)\) \(\approx\) \(1.280604505 + 0.01336219777i\)

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 \( 1 + (0.989 - 0.142i)T \)
11 \( 1 + (-0.415 + 0.909i)T \)
13 \( 1 + (0.989 + 0.142i)T \)
17 \( 1 + (0.755 - 0.654i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 + (-0.540 - 0.841i)T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (0.281 + 0.959i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.989 + 0.142i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (-0.909 + 0.415i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.755 + 0.654i)T \)
79 \( 1 + (0.142 - 0.989i)T \)
83 \( 1 + (0.540 + 0.841i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (0.540 - 0.841i)T \)
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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

−20.82181242502883355020158049456, −20.3579961146030946440138031103, −19.105955388654310843350444038484, −18.58289258680377065154470183149, −17.96629798748447670133069257625, −16.98503720819499229284799746814, −16.37390193720283078172291615921, −15.47328261296332724442402296569, −14.77241793059098440607151138866, −13.88502328993483106051202384755, −13.39733526004718596560525368587, −12.25167577010596109845982115487, −11.60195465842234922090236479617, −10.77119029029795168749675637714, −10.18963281796465412242989741907, −8.96699405777806768930268576244, −8.21643161319150353238322364646, −7.793327441239941482103792109813, −6.51458993224355964596062307512, −5.64334475690543982407274307793, −5.07005175207887021055859383843, −3.80625965262339274401723785835, −3.172435071586022827874846990099, −1.83304676376275618025285767873, −1.0390313817997003356302083476, 0.96146541613848580064131319509, 1.91019045260815500316801042898, 2.93670264223138490869439614171, 4.04712069262345776241420240881, 4.88520704680432864869572925291, 5.560672570592319116624736403098, 6.72104223480194793337742400109, 7.563945366968447431637796011982, 8.16183235670320562528908671422, 9.1797903178832955155895158197, 9.93956143033760021088934913440, 10.89155732074258495678545712394, 11.514605593998120248534187891231, 12.290038055630454092383873257000, 13.27481356084995840993996943712, 13.97343658895120660159125230725, 14.67209531100744060770340300658, 15.602529655642032242442129167039, 16.09421274832706190857904655553, 17.30053977603104376486065910061, 17.73165079102484340273599514223, 18.482291945500621984416477879406, 19.22360360863626876265540548821, 20.407196371227488084406350628209, 20.69893777535222467836974107130

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