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Properties

Label 1-1323-1323.1093-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.817 + 0.576i$
Analytic cond. $6.14398$
Root an. cond. $6.14398$
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  + (0.995 + 0.0995i)2-s + (0.980 + 0.198i)4-s + (0.921 − 0.388i)5-s + (0.955 + 0.294i)8-s + (0.955 − 0.294i)10-s + (−0.583 + 0.811i)11-s + (0.698 + 0.715i)13-s + (0.921 + 0.388i)16-s + (−0.988 − 0.149i)17-s + (−0.5 + 0.866i)19-s + (0.980 − 0.198i)20-s + (−0.661 + 0.749i)22-s + (0.980 + 0.198i)23-s + (0.698 − 0.715i)25-s + (0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (0.995 + 0.0995i)2-s + (0.980 + 0.198i)4-s + (0.921 − 0.388i)5-s + (0.955 + 0.294i)8-s + (0.955 − 0.294i)10-s + (−0.583 + 0.811i)11-s + (0.698 + 0.715i)13-s + (0.921 + 0.388i)16-s + (−0.988 − 0.149i)17-s + (−0.5 + 0.866i)19-s + (0.980 − 0.198i)20-s + (−0.661 + 0.749i)22-s + (0.980 + 0.198i)23-s + (0.698 − 0.715i)25-s + (0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.817 + 0.576i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1093, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.817 + 0.576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.484383159 + 1.104309802i\)
\(L(\frac12)\) \(\approx\) \(3.484383159 + 1.104309802i\)
\(L(1)\) \(\approx\) \(2.249454480 + 0.3305102626i\)
\(L(1)\) \(\approx\) \(2.249454480 + 0.3305102626i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.995 + 0.0995i)T \)
5 \( 1 + (0.921 - 0.388i)T \)
11 \( 1 + (-0.583 + 0.811i)T \)
13 \( 1 + (0.698 + 0.715i)T \)
17 \( 1 + (-0.988 - 0.149i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.980 + 0.198i)T \)
29 \( 1 + (-0.318 + 0.947i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (-0.124 + 0.992i)T \)
43 \( 1 + (0.921 + 0.388i)T \)
47 \( 1 + (-0.411 - 0.911i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (-0.998 - 0.0498i)T \)
61 \( 1 + (0.980 - 0.198i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.365 - 0.930i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.698 - 0.715i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−21.0729908951351994608390955986, −20.44631062785044256235720145868, −19.34023399726663720928993852069, −18.79264389249484125656631337956, −17.63610179445514274908376820500, −17.17439598013864965539336554452, −15.92979489110713366364678595393, −15.49762216878063746828388237119, −14.64513581948158091520231926433, −13.69449033300124292737346302302, −13.33506115276919792616465871924, −12.71879414849404739953395253163, −11.48151094238381656640361762008, −10.746573657972570297376771569378, −10.42000482823042883463846630908, −9.11963055131493231720896161335, −8.26119142824104759428512840018, −7.04104465679188660591091654047, −6.405549781927841715560997256537, −5.63072542522057567869348238186, −4.95850714865191579548610898602, −3.82766037102163814170491163047, −2.84325168615391452853974057220, −2.31408202644370572744556332389, −1.03361109894349321767721591152, 1.51405886080524807532099089003, 2.116122638808706385042498325796, 3.14219785119088157282037821399, 4.33503976143548782078351797890, 4.88279064415763320785630506138, 5.826658255606423983820530990985, 6.5266541316027535650588779415, 7.302845825365805960749751254895, 8.43479495481811731126639957238, 9.283187631910245041339634018160, 10.31328474478605476072634969126, 10.95635264867421301798293932172, 11.946742589410512127465014885003, 12.79381305466289736457863348836, 13.290668421108355645067301573935, 13.97479955464506235738684734314, 14.82259033826458128065352991287, 15.53978405261586416140268687145, 16.40263220795370618817953447750, 17.02423061675934769101415732887, 17.869962334806293910864082942916, 18.7189446571157313500010045088, 19.78857049142537127654079295954, 20.60908659860475586235858610842, 21.05964415324123947680147781826

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