Properties

Label 2-1815-1.1-c1-0-59
Degree $2$
Conductor $1815$
Sign $-1$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 5-s + 7-s + 9-s − 2·12-s − 2·13-s − 15-s + 4·16-s − 6·17-s + 7·19-s + 2·20-s + 21-s − 6·23-s + 25-s + 27-s − 2·28-s − 31-s − 35-s − 2·36-s − 7·37-s − 2·39-s + 6·41-s − 8·43-s − 45-s + 4·48-s − 6·49-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.577·12-s − 0.554·13-s − 0.258·15-s + 16-s − 1.45·17-s + 1.60·19-s + 0.447·20-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.179·31-s − 0.169·35-s − 1/3·36-s − 1.15·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s + 0.577·48-s − 6/7·49-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: yes
Arithmetic: yes
Character: $\chi_{1815} (1, \cdot )$
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 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−8.901173099229814277480267609020, −8.080762717016799727052973771035, −7.60746345376476564860500482297, −6.59523037473790869546358443506, −5.36747921289838822076634820154, −4.65725991292015173841856541050, −3.90838681993568032244631066955, −2.97589293278646371195021990890, −1.63448908436523374581135103796, 0, 1.63448908436523374581135103796, 2.97589293278646371195021990890, 3.90838681993568032244631066955, 4.65725991292015173841856541050, 5.36747921289838822076634820154, 6.59523037473790869546358443506, 7.60746345376476564860500482297, 8.080762717016799727052973771035, 8.901173099229814277480267609020

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