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Properties

Label 1-1375-1375.1113-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.0471 - 0.998i$
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  + (−0.368 − 0.929i)2-s + (0.904 − 0.425i)3-s + (−0.728 + 0.684i)4-s + (−0.728 − 0.684i)6-s + (0.587 − 0.809i)7-s + (0.904 + 0.425i)8-s + (0.637 − 0.770i)9-s + (−0.368 + 0.929i)12-s + (0.770 + 0.637i)13-s + (−0.968 − 0.248i)14-s + (0.0627 − 0.998i)16-s + (0.982 − 0.187i)17-s + (−0.951 − 0.309i)18-s + (0.876 − 0.481i)19-s + (0.187 − 0.982i)21-s + ⋯
L(s)  = 1  + (−0.368 − 0.929i)2-s + (0.904 − 0.425i)3-s + (−0.728 + 0.684i)4-s + (−0.728 − 0.684i)6-s + (0.587 − 0.809i)7-s + (0.904 + 0.425i)8-s + (0.637 − 0.770i)9-s + (−0.368 + 0.929i)12-s + (0.770 + 0.637i)13-s + (−0.968 − 0.248i)14-s + (0.0627 − 0.998i)16-s + (0.982 − 0.187i)17-s + (−0.951 − 0.309i)18-s + (0.876 − 0.481i)19-s + (0.187 − 0.982i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0471 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0471 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.0471 - 0.998i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.0471 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.443015000 - 1.512661316i\)
\(L(\frac12)\) \(\approx\) \(1.443015000 - 1.512661316i\)
\(L(1)\) \(\approx\) \(1.132076974 - 0.7471708058i\)
\(L(1)\) \(\approx\) \(1.132076974 - 0.7471708058i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.368 - 0.929i)T \)
3 \( 1 + (0.904 - 0.425i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (0.770 + 0.637i)T \)
17 \( 1 + (0.982 - 0.187i)T \)
19 \( 1 + (0.876 - 0.481i)T \)
23 \( 1 + (-0.248 + 0.968i)T \)
29 \( 1 + (-0.187 + 0.982i)T \)
31 \( 1 + (0.876 - 0.481i)T \)
37 \( 1 + (-0.368 + 0.929i)T \)
41 \( 1 + (-0.0627 + 0.998i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (0.982 + 0.187i)T \)
53 \( 1 + (0.684 + 0.728i)T \)
59 \( 1 + (-0.0627 + 0.998i)T \)
61 \( 1 + (0.929 - 0.368i)T \)
67 \( 1 + (-0.481 - 0.876i)T \)
71 \( 1 + (0.728 - 0.684i)T \)
73 \( 1 + (-0.770 + 0.637i)T \)
79 \( 1 + (-0.992 + 0.125i)T \)
83 \( 1 + (-0.125 + 0.992i)T \)
89 \( 1 + (0.929 - 0.368i)T \)
97 \( 1 + (-0.904 + 0.425i)T \)
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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

−20.88562787833486995981881285928, −20.3941256769839814103846450322, −19.24674706944594672921328952876, −18.73596612652011114455451854521, −18.08293970368712232497476631720, −17.23316046871620656272238684402, −16.14789243757569219534697596116, −15.792925268303032496539894096779, −14.95573022986534994292135737865, −14.40131379387037973378319004499, −13.756000109664050180363107274226, −12.847182816156944845423866390870, −11.82262522041255896020686433138, −10.56267961116908296234580041174, −10.027102495849269878343369900109, −9.12517592983181247030341737460, −8.32736046353794655579754265267, −8.044104005762787374646829433740, −7.05894374660678232163026637898, −5.81976810589027268191466059752, −5.31664854040629741077127616488, −4.27268304602805650363236691844, −3.377903354871807159242723596720, −2.20117016744569125211885457497, −1.12817437320925536605601174340, 1.15083697688102589632068755710, 1.46266314261493650837788647290, 2.74522218215108677348266627771, 3.51823189824803607141682262935, 4.20985579557826229488914777403, 5.26040844722406226908741896140, 6.780764631097780731083501879590, 7.57690903359681551564891088300, 8.15704598938107026554446480106, 9.013301480787181182724011526987, 9.74386657475096174009639897458, 10.480827217227120702561444785285, 11.54058932134777214571983124388, 11.97475031758736479282379290562, 13.13609164164799454536610692733, 13.743617060030460371492755920242, 14.09110119290794904681028653878, 15.1583947473566995437289385161, 16.26147296570531527254931314831, 17.07041247256232709223256261198, 17.95120723046709125172617099696, 18.510195333153103109724235311958, 19.174500052186071364602033699, 20.099198465400557249321965393358, 20.384890231036409987269341476583

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