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Properties

Label 1-297-297.41-r0-0-0
Degree $1$
Conductor $297$
Sign $0.253 + 0.967i$
Analytic cond. $1.37926$
Root an. cond. $1.37926$
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  + (0.0348 + 0.999i)2-s + (−0.997 + 0.0697i)4-s + (0.882 − 0.469i)5-s + (0.241 + 0.970i)7-s + (−0.104 − 0.994i)8-s + (0.5 + 0.866i)10-s + (0.615 − 0.788i)13-s + (−0.961 + 0.275i)14-s + (0.990 − 0.139i)16-s + (0.669 − 0.743i)17-s + (0.104 + 0.994i)19-s + (−0.848 + 0.529i)20-s + (−0.766 + 0.642i)23-s + (0.559 − 0.829i)25-s + (0.809 + 0.587i)26-s + ⋯
L(s)  = 1  + (0.0348 + 0.999i)2-s + (−0.997 + 0.0697i)4-s + (0.882 − 0.469i)5-s + (0.241 + 0.970i)7-s + (−0.104 − 0.994i)8-s + (0.5 + 0.866i)10-s + (0.615 − 0.788i)13-s + (−0.961 + 0.275i)14-s + (0.990 − 0.139i)16-s + (0.669 − 0.743i)17-s + (0.104 + 0.994i)19-s + (−0.848 + 0.529i)20-s + (−0.766 + 0.642i)23-s + (0.559 − 0.829i)25-s + (0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.253 + 0.967i$
Analytic conductor: \(1.37926\)
Root analytic conductor: \(1.37926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (0:\ ),\ 0.253 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.110285694 + 0.8568472597i\)
\(L(\frac12)\) \(\approx\) \(1.110285694 + 0.8568472597i\)
\(L(1)\) \(\approx\) \(1.053828143 + 0.5501517680i\)
\(L(1)\) \(\approx\) \(1.053828143 + 0.5501517680i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0348 + 0.999i)T \)
5 \( 1 + (0.882 - 0.469i)T \)
7 \( 1 + (0.241 + 0.970i)T \)
13 \( 1 + (0.615 - 0.788i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.961 + 0.275i)T \)
31 \( 1 + (-0.374 - 0.927i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (0.961 - 0.275i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (0.997 + 0.0697i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.438 + 0.898i)T \)
61 \( 1 + (0.374 - 0.927i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (-0.913 + 0.406i)T \)
79 \( 1 + (-0.0348 - 0.999i)T \)
83 \( 1 + (-0.615 - 0.788i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.882 - 0.469i)T \)
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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

−25.50040839570709829260896524050, −23.97904621956507923194363070809, −23.33659399202508318864983881913, −22.30534567391646883428320479052, −21.44847230206168023806580291827, −20.87870909136246689353807815789, −19.84004795616329568384635497615, −18.99667163517788167169766661845, −17.94776494950028039114130070199, −17.40305669574155363454377743546, −16.30100331066532792457352065349, −14.59328405310226091049028572860, −13.98088004840975796020089580112, −13.28336657165807091393449917199, −12.15890391182732657964198155329, −10.92502770568955016396275353839, −10.45660817984800487420413454401, −9.4588501080468824721571504783, −8.42129204513213889331755193177, −7.01572703304691403179504116559, −5.83671196160380651293419580367, −4.547311522652474857114504066293, −3.55392280241478397930438546172, −2.27148986027069905221696075030, −1.18829606942755644738009523353, 1.34019675119338743547833824946, 3.01110680065641109991376240663, 4.573383214321099071175419740650, 5.74647043014041345002954976615, 5.95946647882003738129209642515, 7.62793695074452921168016380796, 8.48182200859249270048456867347, 9.39360411413160407067589591842, 10.22759334923803988632232135158, 11.94677670937447913208498366265, 12.82153326219180426246005843047, 13.78998500031338690954968454957, 14.56239988312267424836960026982, 15.65674618434275448556498054967, 16.34898628474734798559945650211, 17.41722232279043124564679601402, 18.154467680441832463022338769580, 18.795055774537348455083545231087, 20.38627505764019006264597738681, 21.29405970656522377866818768371, 22.09257441484733936214533243260, 22.98386456080185623017182902767, 24.01242409010760670058437126407, 25.0192751720675220444012874320, 25.24136432111061861673016414793

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