Properties

Label 2-1815-1.1-c1-0-13
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 $0$

Origins

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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: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.445939289\)
\(L(\frac12)\) \(\approx\) \(1.445939289\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.93T + 97T^{2} \)
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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

−9.224994025827226373530013479040, −8.346771050865323269795751276658, −7.84605478710161565382743138994, −6.60133654974004880197596118106, −5.94383431608588669416431329664, −5.05449336978319709353830316699, −4.35763858972027751852824011858, −3.76332371883082025486930046560, −2.37145179638812239777919654933, −0.77167416610718969447678728636, 0.77167416610718969447678728636, 2.37145179638812239777919654933, 3.76332371883082025486930046560, 4.35763858972027751852824011858, 5.05449336978319709353830316699, 5.94383431608588669416431329664, 6.60133654974004880197596118106, 7.84605478710161565382743138994, 8.346771050865323269795751276658, 9.224994025827226373530013479040

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