Properties

Label 2-1815-1.1-c1-0-35
Degree $2$
Conductor $1815$
Sign $-1$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.933·2-s − 3-s − 1.12·4-s − 5-s + 0.933·6-s − 2.04·7-s + 2.92·8-s + 9-s + 0.933·10-s + 1.12·12-s − 1.44·13-s + 1.90·14-s + 15-s − 0.469·16-s + 0.867·17-s − 0.933·18-s + 3.12·19-s + 1.12·20-s + 2.04·21-s − 4.70·23-s − 2.92·24-s + 25-s + 1.35·26-s − 27-s + 2.30·28-s + 2.03·29-s − 0.933·30-s + ⋯
L(s)  = 1  − 0.660·2-s − 0.577·3-s − 0.564·4-s − 0.447·5-s + 0.381·6-s − 0.771·7-s + 1.03·8-s + 0.333·9-s + 0.295·10-s + 0.325·12-s − 0.401·13-s + 0.509·14-s + 0.258·15-s − 0.117·16-s + 0.210·17-s − 0.220·18-s + 0.717·19-s + 0.252·20-s + 0.445·21-s − 0.981·23-s − 0.596·24-s + 0.200·25-s + 0.265·26-s − 0.192·27-s + 0.435·28-s + 0.378·29-s − 0.170·30-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + 0.933T + 2T^{2} \)
7 \( 1 + 2.04T + 7T^{2} \)
13 \( 1 + 1.44T + 13T^{2} \)
17 \( 1 - 0.867T + 17T^{2} \)
19 \( 1 - 3.12T + 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 - 2.03T + 29T^{2} \)
31 \( 1 - 10.6T + 31T^{2} \)
37 \( 1 - 4.15T + 37T^{2} \)
41 \( 1 + 0.805T + 41T^{2} \)
43 \( 1 + 2.34T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 - 7.21T + 53T^{2} \)
59 \( 1 + 8.32T + 59T^{2} \)
61 \( 1 + 8.76T + 61T^{2} \)
67 \( 1 + 3.15T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 - 6.14T + 83T^{2} \)
89 \( 1 - 3.77T + 89T^{2} \)
97 \( 1 + 1.93T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.951146869130418726645013793628, −8.069033635736649394560599217722, −7.47017553852012013694285611649, −6.55721578720501856854120829398, −5.68011195973462152573481184749, −4.69331972092302696543737130346, −3.99715555321544221540555589964, −2.81677873845107551796920597204, −1.14544647835093367988417822886, 0, 1.14544647835093367988417822886, 2.81677873845107551796920597204, 3.99715555321544221540555589964, 4.69331972092302696543737130346, 5.68011195973462152573481184749, 6.55721578720501856854120829398, 7.47017553852012013694285611649, 8.069033635736649394560599217722, 8.951146869130418726645013793628

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