Properties

Label 1-75-75.47-r0-0-0
Degree $1$
Conductor $75$
Sign $0.0627 + 0.998i$
Analytic cond. $0.348298$
Root an. cond. $0.348298$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + i·7-s + (0.951 + 0.309i)8-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + (−0.309 + 0.951i)19-s + (−0.951 + 0.309i)22-s + (0.587 − 0.809i)23-s − 26-s + (0.951 − 0.309i)28-s + (0.309 + 0.951i)29-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + i·7-s + (0.951 + 0.309i)8-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + (−0.309 + 0.951i)19-s + (−0.951 + 0.309i)22-s + (0.587 − 0.809i)23-s − 26-s + (0.951 − 0.309i)28-s + (0.309 + 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.0627 + 0.998i$
Analytic conductor: \(0.348298\)
Root analytic conductor: \(0.348298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 75,\ (0:\ ),\ 0.0627 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5145343640 + 0.4831799291i\)
\(L(\frac12)\) \(\approx\) \(0.5145343640 + 0.4831799291i\)
\(L(1)\) \(\approx\) \(0.6932807922 + 0.3694459456i\)
\(L(1)\) \(\approx\) \(0.6932807922 + 0.3694459456i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−30.68495358895928416643905682306, −30.04268027082378850564457882523, −29.08081426458658453574541189013, −27.884902913961841056326606630769, −26.980028403225898633225631398315, −26.10142858611440569873126615018, −24.84589908600967977785192120227, −23.33570442229794829252824107145, −22.22232590118795008190690822812, −21.10966178439793814638227143890, −19.95359543190405934365487153221, −19.30668599597741459634699501058, −17.73633069050242069500722042726, −17.092103334223862247692956266020, −15.69974537225162672503834757936, −13.82877756948052924151945976671, −12.98724911848194310319231932823, −11.39230025396753234169376961374, −10.63409432877592657340149197485, −9.25567200052589881779386892556, −8.07761255540006043111152582605, −6.64928265015925727443547785805, −4.40669136417953486205794633918, −3.14146748743756493496022343733, −1.13934320330896578457397579335, 1.91283244724449101647553162245, 4.40758427749293681239107420509, 5.94273778665780996990159046617, 6.984751490268936761294210346762, 8.630406500912311385739448879730, 9.32462154170658071698359413951, 10.8986318651360808969119014877, 12.33364092889081157504954928205, 14.00752377383222957016974825327, 15.025197036788251030275504692972, 16.05492582451063414353990708955, 17.19405298847476545129989753259, 18.36099548630585810681062496934, 19.16041998469709582607862233335, 20.53762455575362890737584494707, 22.09348670055455990524461677147, 23.09453844692493047855105704881, 24.4311247494905800393045550540, 25.151588135248619447061964595779, 26.15981693098927996019430002838, 27.36445660857412007745060637390, 28.2305296853502028824106344903, 29.13904921986625520861429443954, 30.87705506003229924395897994776, 31.74164386975055366343991095650

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