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Properties

Label 1-15-15.8-r0-0-0
Degree $1$
Conductor $15$
Sign $0.525 + 0.850i$
Analytic cond. $0.0696597$
Root an. cond. $0.0696597$
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  + i·2-s − 4-s − i·7-s − i·8-s − 11-s + i·13-s + 14-s + 16-s + i·17-s − 19-s − i·22-s − i·23-s − 26-s + i·28-s + 29-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·7-s − i·8-s − 11-s + i·13-s + 14-s + 16-s + i·17-s − 19-s − i·22-s − i·23-s − 26-s + i·28-s + 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(0.0696597\)
Root analytic conductor: \(0.0696597\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 15,\ (0:\ ),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4818861612 + 0.2686691313i\)
\(L(\frac12)\) \(\approx\) \(0.4818861612 + 0.2686691313i\)
\(L(1)\) \(\approx\) \(0.7385666321 + 0.3168026445i\)
\(L(1)\) \(\approx\) \(0.7385666321 + 0.3168026445i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 - iT \)
23 \( 1 - iT \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 - T \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 + T \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 - iT \)
89 \( 1 - iT \)
97 \( 1 \)
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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

−42.12516342957911714324660084166, −40.86973749582400795802655221661, −39.61717684267314095808062933406, −38.27968410596217858204325919302, −37.22297310571034111855618795304, −35.8198791029811636818283511977, −34.25977663837223024363887134687, −32.158490906683784062201633697714, −31.19235656367538178255967339596, −29.66537127834534020045555129427, −28.39897714383705927587701549747, −27.1947316268971750189066901495, −25.38802788017747656542479000231, −23.3384800177598776614370034132, −21.88397302420891165032133756714, −20.62763177596965030615094928929, −18.97699965008825353453361850360, −17.77239024921666406085023142599, −15.36613957649778231479471833209, −13.33739585541558924038778494384, −11.90737247654667032066686916102, −10.18760272764530244103922817377, −8.414685249804899271264541932753, −5.24301049275448939295780629412, −2.73460370911883725526846548734, 4.40670023980367373719937803757, 6.59078267021547088948133257241, 8.26450165390342495596227258835, 10.326204250314941250444125974117, 13.02895627399163930052928661250, 14.46446113446334317877184065112, 16.17189288498984240741590706321, 17.44476426609567023348344086788, 19.13784242419278062325841722270, 21.28464369710184872705959727366, 23.14894539492477018901287021167, 24.07185831517868257832906114818, 25.892588846441112172235181503, 26.79623559700210519510149195478, 28.55271652717981339034779752589, 30.51934589147356387988102083237, 31.99774120359653597669262975132, 33.29702889306197928384987245938, 34.379512734862475731909907621677, 35.991184135736971037638640809770, 36.85300610546312603666526988175, 39.02994490198785725513290035559, 40.40724799364991335221418635001, 41.808349985014165999233200066486, 42.887072627995326261616085091056

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