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Properties

Label 1-2541-2541.5-r0-0-0
Degree $1$
Conductor $2541$
Sign $-0.959 - 0.281i$
Analytic cond. $11.8003$
Root an. cond. $11.8003$
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  + (0.532 − 0.846i)2-s + (−0.432 − 0.901i)4-s + (−0.948 + 0.318i)5-s + (−0.993 − 0.113i)8-s + (−0.235 + 0.971i)10-s + (−0.941 + 0.336i)13-s + (−0.625 + 0.780i)16-s + (−0.290 + 0.956i)17-s + (0.999 + 0.0190i)19-s + (0.696 + 0.717i)20-s + (−0.0475 + 0.998i)23-s + (0.797 − 0.603i)25-s + (−0.217 + 0.976i)26-s + (0.921 − 0.389i)29-s + (−0.380 − 0.924i)31-s + (0.327 + 0.945i)32-s + ⋯
L(s)  = 1  + (0.532 − 0.846i)2-s + (−0.432 − 0.901i)4-s + (−0.948 + 0.318i)5-s + (−0.993 − 0.113i)8-s + (−0.235 + 0.971i)10-s + (−0.941 + 0.336i)13-s + (−0.625 + 0.780i)16-s + (−0.290 + 0.956i)17-s + (0.999 + 0.0190i)19-s + (0.696 + 0.717i)20-s + (−0.0475 + 0.998i)23-s + (0.797 − 0.603i)25-s + (−0.217 + 0.976i)26-s + (0.921 − 0.389i)29-s + (−0.380 − 0.924i)31-s + (0.327 + 0.945i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $-0.959 - 0.281i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(11.8003\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2541,\ (0:\ ),\ -0.959 - 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1198388820 - 0.8346659606i\)
\(L(\frac12)\) \(\approx\) \(0.1198388820 - 0.8346659606i\)
\(L(1)\) \(\approx\) \(0.8236367938 - 0.4504954701i\)
\(L(1)\) \(\approx\) \(0.8236367938 - 0.4504954701i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.532 - 0.846i)T \)
5 \( 1 + (-0.948 + 0.318i)T \)
13 \( 1 + (-0.941 + 0.336i)T \)
17 \( 1 + (-0.290 + 0.956i)T \)
19 \( 1 + (0.999 + 0.0190i)T \)
23 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (0.921 - 0.389i)T \)
31 \( 1 + (-0.380 - 0.924i)T \)
37 \( 1 + (0.879 - 0.475i)T \)
41 \( 1 + (-0.985 + 0.170i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (-0.830 - 0.556i)T \)
53 \( 1 + (0.625 + 0.780i)T \)
59 \( 1 + (0.640 - 0.768i)T \)
61 \( 1 + (-0.999 + 0.0380i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (-0.0855 - 0.996i)T \)
73 \( 1 + (-0.161 - 0.986i)T \)
79 \( 1 + (0.00951 + 0.999i)T \)
83 \( 1 + (-0.998 - 0.0570i)T \)
89 \( 1 + (0.981 + 0.189i)T \)
97 \( 1 + (-0.198 - 0.980i)T \)
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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

−19.97996853494888727270106045457, −18.950545692225096950329592792326, −18.11236058593956259440036158067, −17.59778775868354283615089571771, −16.472640795600999271854177256530, −16.26883743597069350009936920142, −15.52993195968037070079920934873, −14.73706046548756385911025754137, −14.26739080992215820261516820710, −13.30907186510915055057580868227, −12.62743307992345952148875634702, −11.94256729267318009308595123396, −11.47393813378998615094441566078, −10.26645160719052629119121807792, −9.34278240060334886226705237733, −8.57901959007189051427791437625, −7.90525175218033444220268234144, −7.1909815887554230259796626671, −6.67560085712079942844824385012, −5.47517446766468572932067079527, −4.82649223455171849320519662099, −4.32895715484892948764713997040, −3.18353846492506778103550761115, −2.71454022318825101814165572831, −0.95596290076159989020177960516, 0.27093511663118577720189124117, 1.51480889133685542737368855466, 2.4535131434412038630311637612, 3.29124591436072600780166678284, 3.98811801977697536575445527649, 4.68065116092599730719801952344, 5.53898515018093544960825908204, 6.4490325295063884537163274803, 7.34577424821122491828006932341, 8.09059790651240360408939096580, 9.09474233705614408117847974380, 9.81658112694126748135563508600, 10.55862650735393002067276562072, 11.330686558636080453958967566702, 11.90834875391077445528250734808, 12.40119526305689431903530405725, 13.36011043786555915041125897042, 13.981311041362703522509295425836, 14.93524062237563383128605363493, 15.187416152543540839351044651550, 16.08383146068751620351177503539, 17.0270842099765626251831961830, 17.91722642531383167710531069831, 18.60757931231707761303578211265, 19.35654086540759628349855430890

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