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Properties

Label 1-297-297.65-r0-0-0
Degree $1$
Conductor $297$
Sign $-0.597 + 0.802i$
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  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s + (0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)10-s + (−0.173 + 0.984i)13-s + (−0.173 + 0.984i)14-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.939 + 0.342i)20-s + (0.939 − 0.342i)23-s + (0.173 + 0.984i)25-s − 26-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s + (0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)10-s + (−0.173 + 0.984i)13-s + (−0.173 + 0.984i)14-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.939 + 0.342i)20-s + (0.939 − 0.342i)23-s + (0.173 + 0.984i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.597 + 0.802i$
Analytic conductor: \(1.37926\)
Root analytic conductor: \(1.37926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (0:\ ),\ -0.597 + 0.802i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4587163435 + 0.9133793361i\)
\(L(\frac12)\) \(\approx\) \(0.4587163435 + 0.9133793361i\)
\(L(1)\) \(\approx\) \(0.7862892570 + 0.5574088528i\)
\(L(1)\) \(\approx\) \(0.7862892570 + 0.5574088528i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−24.942475812422333777386371452742, −23.90051298189912970964429771449, −23.10297666132545233804484754420, −22.394071188995239089456155380, −21.48428485035038936728090187516, −20.36461427089637121423157470849, −19.92528354970941096919080823599, −18.82230873815305724202806738837, −18.01427996653594824516352891152, −17.28946959675244297927268660909, −15.61630340935116848636812582007, −14.85477918390741769284416271257, −13.925192947954114023880878890196, −12.99086327183323970153073485521, −11.73118720968091551826244127909, −11.20413359620663629663097227810, −10.41519701414894583971718985799, −9.1827281728133724954207565983, −8.02727321008162586812390210996, −7.13368156040854333686094382334, −5.38418621080416645722564179564, −4.48390146212433485181718626682, −3.35563978747644774682909133895, −2.36068897064143373541100244518, −0.70281614558089764944720172950, 1.505858908910709867266035287711, 3.57985533770492518538028945752, 4.6049839533594278015772392589, 5.31801968723221122855280933499, 6.68346870101463012463465921942, 7.71536067525408893742604113286, 8.549132348711030899470248993213, 9.23018636942288015202564249018, 10.88892664285987735853563090209, 12.04212790535232251989863933331, 12.74418840651999564712199014853, 14.036139659094455147364576234731, 14.79529524742225068831699467991, 15.63646348456322090729692029193, 16.57713703954110388581290786073, 17.2481180094516075743041698535, 18.38451907650651225187612581544, 19.15926704858582005357924122547, 20.44580358176200251427464974714, 21.36492179287009865893482224764, 22.24643656334935931497391425503, 23.43247781038244029755205370806, 23.98598102969278470174861765992, 24.61200505808433923366764040592, 25.544687774826497924409426140630

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