Properties

Label 2-405-1.1-c3-0-31
Degree $2$
Conductor $405$
Sign $-1$
Analytic cond. $23.8957$
Root an. cond. $4.88833$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 17·4-s + 5·5-s + 9·7-s − 45·8-s − 25·10-s − 8·11-s + 43·13-s − 45·14-s + 89·16-s − 122·17-s − 59·19-s + 85·20-s + 40·22-s − 213·23-s + 25·25-s − 215·26-s + 153·28-s + 224·29-s − 36·31-s − 85·32-s + 610·34-s + 45·35-s + 206·37-s + 295·38-s − 225·40-s + 413·41-s + ⋯
L(s)  = 1  − 1.76·2-s + 17/8·4-s + 0.447·5-s + 0.485·7-s − 1.98·8-s − 0.790·10-s − 0.219·11-s + 0.917·13-s − 0.859·14-s + 1.39·16-s − 1.74·17-s − 0.712·19-s + 0.950·20-s + 0.387·22-s − 1.93·23-s + 1/5·25-s − 1.62·26-s + 1.03·28-s + 1.43·29-s − 0.208·31-s − 0.469·32-s + 3.07·34-s + 0.217·35-s + 0.915·37-s + 1.25·38-s − 0.889·40-s + 1.57·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-1$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - p T \)
good2 \( 1 + 5 T + p^{3} T^{2} \)
7 \( 1 - 9 T + p^{3} T^{2} \)
11 \( 1 + 8 T + p^{3} T^{2} \)
13 \( 1 - 43 T + p^{3} T^{2} \)
17 \( 1 + 122 T + p^{3} T^{2} \)
19 \( 1 + 59 T + p^{3} T^{2} \)
23 \( 1 + 213 T + p^{3} T^{2} \)
29 \( 1 - 224 T + p^{3} T^{2} \)
31 \( 1 + 36 T + p^{3} T^{2} \)
37 \( 1 - 206 T + p^{3} T^{2} \)
41 \( 1 - 413 T + p^{3} T^{2} \)
43 \( 1 + 392 T + p^{3} T^{2} \)
47 \( 1 + 311 T + p^{3} T^{2} \)
53 \( 1 + 377 T + p^{3} T^{2} \)
59 \( 1 - 337 T + p^{3} T^{2} \)
61 \( 1 - 40 T + p^{3} T^{2} \)
67 \( 1 - 348 T + p^{3} T^{2} \)
71 \( 1 - 62 T + p^{3} T^{2} \)
73 \( 1 + 1214 T + p^{3} T^{2} \)
79 \( 1 + 294 T + p^{3} T^{2} \)
83 \( 1 - 534 T + p^{3} T^{2} \)
89 \( 1 + 810 T + p^{3} T^{2} \)
97 \( 1 + 928 T + p^{3} T^{2} \)
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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

−10.25280996509978192979897566852, −9.431799958851675058299078897424, −8.444616650609701272669961206285, −8.094527534605872981674135545363, −6.72010238527007024588745592357, −6.10893329473747138912572802865, −4.38982254190662338155085937462, −2.47537997320282478561694825160, −1.53045035386671870529241758797, 0, 1.53045035386671870529241758797, 2.47537997320282478561694825160, 4.38982254190662338155085937462, 6.10893329473747138912572802865, 6.72010238527007024588745592357, 8.094527534605872981674135545363, 8.444616650609701272669961206285, 9.431799958851675058299078897424, 10.25280996509978192979897566852

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