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Properties

Label 2-1815-5.4-c1-0-61
Degree $2$
Conductor $1815$
Sign $0.447 - 0.894i$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.456i·2-s + i·3-s + 1.79·4-s + (2 + i)5-s − 0.456·6-s − 0.913i·7-s + 1.73i·8-s − 9-s + (−0.456 + 0.913i)10-s + 1.79i·12-s − 0.913i·13-s + 0.417·14-s + (−1 + 2i)15-s + 2.79·16-s − 7.02i·17-s − 0.456i·18-s + ⋯
L(s)  = 1  + 0.323i·2-s + 0.577i·3-s + 0.895·4-s + (0.894 + 0.447i)5-s − 0.186·6-s − 0.345i·7-s + 0.612i·8-s − 0.333·9-s + (−0.144 + 0.288i)10-s + 0.517i·12-s − 0.253i·13-s + 0.111·14-s + (−0.258 + 0.516i)15-s + 0.697·16-s − 1.70i·17-s − 0.107i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(14.4928\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.724278416\)
\(L(\frac12)\) \(\approx\) \(2.724278416\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (-2 - i)T \)
11 \( 1 \)
good2 \( 1 - 0.456iT - 2T^{2} \)
7 \( 1 + 0.913iT - 7T^{2} \)
13 \( 1 + 0.913iT - 13T^{2} \)
17 \( 1 + 7.02iT - 17T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 - 0.582iT - 23T^{2} \)
29 \( 1 - 4.37T + 29T^{2} \)
31 \( 1 - 2.58T + 31T^{2} \)
37 \( 1 - 9.58iT - 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - 4.37iT - 43T^{2} \)
47 \( 1 - 6.58iT - 47T^{2} \)
53 \( 1 + 5iT - 53T^{2} \)
59 \( 1 - 3.58T + 59T^{2} \)
61 \( 1 + 8.66T + 61T^{2} \)
67 \( 1 + 14.7iT - 67T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 + 2.64T + 79T^{2} \)
83 \( 1 + 12.2iT - 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + 3.58iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.717645170811827020864749715881, −8.626838731611463273240392155598, −7.63121544088818306858630908927, −6.98089220495134057833474164195, −6.26767298444386626755713965107, −5.39453800472160066098542971493, −4.75340786215508608402629700593, −3.12166327638125273884574956941, −2.77909861501784555796056158183, −1.32039125196625985459173677274, 1.18215376249600462972180593677, 1.96721405841818521335009157482, 2.82042686435916883780366075704, 3.97568904475397519861088559243, 5.37478443729035241461726749222, 5.95086539644050682868642135684, 6.66440569707014659154947919736, 7.46456574971493568839437596531, 8.416508562313465728515135058479, 9.064714067095344407578667682344

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