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Properties

Label 1-1380-1380.1307-r0-0-0
Degree $1$
Conductor $1380$
Sign $-0.536 + 0.843i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.536 + 0.843i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ -0.536 + 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3242805609 + 0.5908228902i\)
\(L(\frac12)\) \(\approx\) \(0.3242805609 + 0.5908228902i\)
\(L(1)\) \(\approx\) \(0.8528186759 + 0.09642446031i\)
\(L(1)\) \(\approx\) \(0.8528186759 + 0.09642446031i\)

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

−20.52938046152911751330901983893, −19.60275945050811672194759222684, −19.30279174730847517745161041943, −18.15802402146613287840320190927, −17.733341102285224520267417073857, −16.755064015250763081615188828505, −15.95797914950362629932653707923, −15.2107307699625800207361716065, −14.70268937284150716195717298199, −13.446871958831595385224082521917, −13.08634987903118131476796152569, −11.97483624052066052503170051917, −11.462212327060055090508774171991, −10.534978606157415201478231459052, −9.58181516371484588721482555207, −8.81255457192742874145354516848, −8.26211357136604701677328712841, −7.05700027587273946720845696178, −6.289712566376340335042727864165, −5.50073633864347586171390915992, −4.65926855450358723192385414780, −3.44384921206319106698051708401, −2.75549654545375894836698009172, −1.70868087803011913287900103267, −0.25005836339314087250293822747, 1.38357872406886362826727028964, 2.16294969084527620945608569719, 3.58951397586531587434778065803, 4.14240859076830613702146229666, 5.02093001955060291346009342775, 6.22678973482412165285039345180, 7.03501343591887312320075177458, 7.46331420698104230586551300746, 8.792697511835945964194420912167, 9.39365585157822048598670455333, 10.27264791420825345207696007986, 10.97746339771797481781565546670, 11.944996818971810925796462239161, 12.58700755983576224386415995149, 13.56852492310482257351616384983, 14.18899177581601712514948982525, 14.884268856509992964347365780598, 15.89168255454883794990897504942, 16.661468281241063761319024879727, 17.18354261431238125940875015208, 18.01291235570094830415306552671, 18.91442683808757792413835650533, 19.66691939798458509870447111856, 20.31822895956426689508405846481, 20.885633036894267981102057722412

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