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Properties

Label 1-1375-1375.1317-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.994 + 0.106i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (−0.904 + 0.425i)2-s + (−0.368 + 0.929i)3-s + (0.637 − 0.770i)4-s + (−0.0627 − 0.998i)6-s + (0.587 + 0.809i)7-s + (−0.248 + 0.968i)8-s + (−0.728 − 0.684i)9-s + (0.481 + 0.876i)12-s + (−0.684 + 0.728i)13-s + (−0.876 − 0.481i)14-s + (−0.187 − 0.982i)16-s + (0.368 + 0.929i)17-s + (0.951 + 0.309i)18-s + (0.0627 + 0.998i)19-s + (−0.968 + 0.248i)21-s + ⋯
L(s)  = 1  + (−0.904 + 0.425i)2-s + (−0.368 + 0.929i)3-s + (0.637 − 0.770i)4-s + (−0.0627 − 0.998i)6-s + (0.587 + 0.809i)7-s + (−0.248 + 0.968i)8-s + (−0.728 − 0.684i)9-s + (0.481 + 0.876i)12-s + (−0.684 + 0.728i)13-s + (−0.876 − 0.481i)14-s + (−0.187 − 0.982i)16-s + (0.368 + 0.929i)17-s + (0.951 + 0.309i)18-s + (0.0627 + 0.998i)19-s + (−0.968 + 0.248i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.994 + 0.106i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.994 + 0.106i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04340232815 + 0.8150968412i\)
\(L(\frac12)\) \(\approx\) \(0.04340232815 + 0.8150968412i\)
\(L(1)\) \(\approx\) \(0.5052245106 + 0.4463303007i\)
\(L(1)\) \(\approx\) \(0.5052245106 + 0.4463303007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.904 + 0.425i)T \)
3 \( 1 + (-0.368 + 0.929i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (-0.684 + 0.728i)T \)
17 \( 1 + (0.368 + 0.929i)T \)
19 \( 1 + (0.0627 + 0.998i)T \)
23 \( 1 + (0.982 + 0.187i)T \)
29 \( 1 + (0.535 + 0.844i)T \)
31 \( 1 + (0.968 + 0.248i)T \)
37 \( 1 + (-0.684 + 0.728i)T \)
41 \( 1 + (-0.728 - 0.684i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (0.998 + 0.0627i)T \)
53 \( 1 + (-0.998 - 0.0627i)T \)
59 \( 1 + (0.992 + 0.125i)T \)
61 \( 1 + (0.992 - 0.125i)T \)
67 \( 1 + (-0.368 - 0.929i)T \)
71 \( 1 + (-0.929 - 0.368i)T \)
73 \( 1 + (0.904 - 0.425i)T \)
79 \( 1 + (0.968 - 0.248i)T \)
83 \( 1 + (-0.248 + 0.968i)T \)
89 \( 1 + (-0.876 - 0.481i)T \)
97 \( 1 + (-0.770 - 0.637i)T \)
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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

−20.39339153778987286843030308149, −19.46271845997158009023442179520, −19.10336790509335807231914414012, −18.041047003239984007159034100973, −17.54283180742567824956333690103, −17.11767795803644552172786138531, −16.251554417809612661895333730804, −15.283420031216138500598712058545, −14.1925587143548470460675560723, −13.36944426027233488616449407058, −12.68849075124514842854005496402, −11.772575988268566268033339643708, −11.272535343831801258509844085306, −10.47737533465400222576834703426, −9.684506856517735976055153856111, −8.59214638680627004042563797867, −7.85132681972253569473775464747, −7.22907360693206351829043345546, −6.65782379840544413833132655420, −5.36973199367487903454000985596, −4.46039718243703682025341175731, −2.99161277553377745886189701186, −2.39947056730394438318114175356, −1.13802141110081667783456602981, −0.54168508653314778878266868751, 1.267442767523547639078497829345, 2.29798415451044253965198188949, 3.43908241205842237903709651098, 4.73863557535526546904264053355, 5.342161600544173889153363319435, 6.15084926437680430158241678338, 7.0264612747049398213372418689, 8.19345965637561551303383969163, 8.74964425629800548308096610278, 9.498704931539088731228136144677, 10.31443233889618431592968646622, 10.887060248669508209535050760829, 11.91090322355899377144533657326, 12.25035771409026841835952231141, 14.07555989300269552904717766703, 14.62226398424451708336356852907, 15.31684489090519370319747085523, 15.88593946744805340554632501162, 16.915768531190594961234432468260, 17.15651594354950730404919020648, 18.045362533958506410421398963436, 18.94733109468787077109012851565, 19.4283582499993629256974009162, 20.68629952826726144430336561779, 20.993716363520212230908556082586

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