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Properties

Label 1-675-675.4-r0-0-0
Degree $1$
Conductor $675$
Sign $-0.691 + 0.722i$
Analytic cond. $3.13468$
Root an. cond. $3.13468$
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  + (0.241 + 0.970i)2-s + (−0.882 + 0.469i)4-s + (−0.173 + 0.984i)7-s + (−0.669 − 0.743i)8-s + (0.961 + 0.275i)11-s + (0.241 − 0.970i)13-s + (−0.997 + 0.0697i)14-s + (0.559 − 0.829i)16-s + (0.978 − 0.207i)17-s + (0.669 + 0.743i)19-s + (−0.0348 + 0.999i)22-s + (−0.438 + 0.898i)23-s + 26-s + (−0.309 − 0.951i)28-s + (−0.374 + 0.927i)29-s + ⋯
L(s)  = 1  + (0.241 + 0.970i)2-s + (−0.882 + 0.469i)4-s + (−0.173 + 0.984i)7-s + (−0.669 − 0.743i)8-s + (0.961 + 0.275i)11-s + (0.241 − 0.970i)13-s + (−0.997 + 0.0697i)14-s + (0.559 − 0.829i)16-s + (0.978 − 0.207i)17-s + (0.669 + 0.743i)19-s + (−0.0348 + 0.999i)22-s + (−0.438 + 0.898i)23-s + 26-s + (−0.309 − 0.951i)28-s + (−0.374 + 0.927i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ -0.691 + 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5693526834 + 1.332679733i\)
\(L(\frac12)\) \(\approx\) \(0.5693526834 + 1.332679733i\)
\(L(1)\) \(\approx\) \(0.8828367668 + 0.7311362988i\)
\(L(1)\) \(\approx\) \(0.8828367668 + 0.7311362988i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.241 + 0.970i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (0.961 + 0.275i)T \)
13 \( 1 + (0.241 - 0.970i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.438 + 0.898i)T \)
29 \( 1 + (-0.374 + 0.927i)T \)
31 \( 1 + (0.990 + 0.139i)T \)
37 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + (-0.241 + 0.970i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.990 + 0.139i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.961 - 0.275i)T \)
61 \( 1 + (-0.719 + 0.694i)T \)
67 \( 1 + (0.374 + 0.927i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (-0.913 + 0.406i)T \)
79 \( 1 + (-0.374 + 0.927i)T \)
83 \( 1 + (-0.848 + 0.529i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (0.615 + 0.788i)T \)
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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

−22.49853646885682296602496463655, −21.52474749314290888500401993850, −20.78626680242355677632997812733, −20.07210800723914822631389928930, −19.22239718575130989727585375737, −18.75261789385472346885006925183, −17.47999719641905474146817474595, −16.93297571725230783297044715116, −15.9004198500072330445069119152, −14.56061809531193370980325864685, −13.98185454809001089687238535049, −13.39107761497923363201114501006, −12.212590357418913226092631502075, −11.61503811589897436881598006850, −10.72509041995448122313229366883, −9.86657609949778347344351781068, −9.152994659234344430933548232902, −8.10216439299614947424640781984, −6.85894593833449218302623871497, −5.95724519996874636288193566530, −4.633098840496998102714811588, −3.958227751892062971977929357969, −3.090856965838864914021853072283, −1.73958921014235176087026175648, −0.7621096218954281049621343822, 1.32852527087067351998738612627, 3.05083294956634430954278877069, 3.77716376950337586597928024651, 5.18614182228915176456145015603, 5.7019241603074069177018833165, 6.64020690654913950177087407619, 7.68172648525151024640043676379, 8.435865951979994867893874906304, 9.3926208796139677111880111586, 10.05696389764075314271618087711, 11.69246400069610211483277217579, 12.290205185410042821332656290635, 13.125105730884579313765163009223, 14.23812263557456787347378840214, 14.748183348613528678665252657679, 15.71943025776591632232224104658, 16.24051475007954898274224072567, 17.30882254666536627358504051381, 17.982685327569731472461381588598, 18.74930819020064084533423688664, 19.65508632695789763881088677614, 20.83777068397822352612065433736, 21.63039217828191271015339533726, 22.63331626433459321012545354951, 22.7742211900574747759077493303

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