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Properties

Label 1-3525-3525.548-r1-0-0
Degree $1$
Conductor $3525$
Sign $0.345 - 0.938i$
Analytic cond. $378.813$
Root an. cond. $378.813$
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  + (0.163 + 0.986i)2-s + (−0.946 + 0.321i)4-s + (0.519 − 0.854i)7-s + (−0.472 − 0.881i)8-s + (0.0409 − 0.999i)11-s + (0.496 + 0.868i)13-s + (0.927 + 0.373i)14-s + (0.792 − 0.609i)16-s + (−0.347 − 0.937i)17-s + (0.507 − 0.861i)19-s + (0.992 − 0.122i)22-s + (0.711 − 0.702i)23-s + (−0.775 + 0.631i)26-s + (−0.216 + 0.976i)28-s + (0.740 − 0.672i)29-s + ⋯
L(s)  = 1  + (0.163 + 0.986i)2-s + (−0.946 + 0.321i)4-s + (0.519 − 0.854i)7-s + (−0.472 − 0.881i)8-s + (0.0409 − 0.999i)11-s + (0.496 + 0.868i)13-s + (0.927 + 0.373i)14-s + (0.792 − 0.609i)16-s + (−0.347 − 0.937i)17-s + (0.507 − 0.861i)19-s + (0.992 − 0.122i)22-s + (0.711 − 0.702i)23-s + (−0.775 + 0.631i)26-s + (−0.216 + 0.976i)28-s + (0.740 − 0.672i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.345 - 0.938i$
Analytic conductor: \(378.813\)
Root analytic conductor: \(378.813\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (548, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3525,\ (1:\ ),\ 0.345 - 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.927763985 - 1.344371463i\)
\(L(\frac12)\) \(\approx\) \(1.927763985 - 1.344371463i\)
\(L(1)\) \(\approx\) \(1.161086512 + 0.1369014919i\)
\(L(1)\) \(\approx\) \(1.161086512 + 0.1369014919i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.163 + 0.986i)T \)
7 \( 1 + (0.519 - 0.854i)T \)
11 \( 1 + (0.0409 - 0.999i)T \)
13 \( 1 + (0.496 + 0.868i)T \)
17 \( 1 + (-0.347 - 0.937i)T \)
19 \( 1 + (0.507 - 0.861i)T \)
23 \( 1 + (0.711 - 0.702i)T \)
29 \( 1 + (0.740 - 0.672i)T \)
31 \( 1 + (-0.824 - 0.565i)T \)
37 \( 1 + (-0.373 - 0.927i)T \)
41 \( 1 + (-0.435 - 0.900i)T \)
43 \( 1 + (0.816 + 0.576i)T \)
53 \( 1 + (0.472 - 0.881i)T \)
59 \( 1 + (0.598 - 0.800i)T \)
61 \( 1 + (0.927 + 0.373i)T \)
67 \( 1 + (-0.922 - 0.385i)T \)
71 \( 1 + (0.894 - 0.447i)T \)
73 \( 1 + (0.932 - 0.360i)T \)
79 \( 1 + (0.981 - 0.190i)T \)
83 \( 1 + (0.784 + 0.620i)T \)
89 \( 1 + (-0.662 + 0.749i)T \)
97 \( 1 + (-0.565 - 0.824i)T \)
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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

−18.71108524688431656740751751799, −17.99786481103001229682328418793, −17.72600301390260807570239036404, −16.86722340378487931017772359165, −15.678674288847433534412544860838, −15.0981892643379634514280929133, −14.592760981908662020775231351663, −13.73742831380125426468217113946, −12.90518599928589564128800427443, −12.41888106534450930154158966085, −11.842302044350713159308517649935, −11.00983752002205213819369395226, −10.41171128111997181344268195173, −9.724536222611552035563143173786, −8.86902853730753532286447708394, −8.35886083122381355680972488113, −7.54357645131614733211463909532, −6.392905632612469860095825836362, −5.44834230493195164377819107789, −5.09386236583055743224491963914, −4.092581697596034853787065216012, −3.33548808067465037533588925185, −2.53791961727915139160976266272, −1.637108148515571846036999499149, −1.13452446019559687326971568263, 0.43884570828865553926416352438, 0.85340249017680570625021947306, 2.2889726113701570672739274546, 3.41291249035745854892646383735, 4.073898072370642478212341903340, 4.8354678068051473860980177608, 5.46990909633236840374195949935, 6.49692818217710066002368611238, 6.96113034047019198649011165821, 7.67503471404376599017191227693, 8.51683450330454663715362438540, 9.0529094033705206869329771066, 9.785058129350555038417051183579, 10.95091129887880899643022697635, 11.290463761465410941471417625408, 12.29210099381542229437647389580, 13.300335806733706336403949636626, 13.77161498197291902296453696970, 14.145471814067873037569499663294, 14.97691032256072187727917658819, 15.834896467331649690636014126181, 16.39117890072413443247540127614, 16.83539422920172773702742993785, 17.7524998033047494872277460192, 18.13458343830350030586314255532

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