Properties

Label 2-15e2-1.1-c7-0-37
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $70.2866$
Root an. cond. $8.38371$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22·2-s + 356·4-s + 420·7-s − 5.01e3·8-s + 2.94e3·11-s + 1.10e4·13-s − 9.24e3·14-s + 6.47e4·16-s − 1.65e4·17-s − 2.53e4·19-s − 6.47e4·22-s − 5.88e3·23-s − 2.42e5·26-s + 1.49e5·28-s − 1.63e5·29-s − 2.01e5·31-s − 7.83e5·32-s + 3.64e5·34-s − 1.20e5·37-s + 5.58e5·38-s + 1.15e5·41-s + 1.91e4·43-s + 1.04e6·44-s + 1.29e5·46-s + 8.41e5·47-s − 6.47e5·49-s + 3.91e6·52-s + ⋯
L(s)  = 1  − 1.94·2-s + 2.78·4-s + 0.462·7-s − 3.46·8-s + 0.666·11-s + 1.38·13-s − 0.899·14-s + 3.95·16-s − 0.816·17-s − 0.848·19-s − 1.29·22-s − 0.100·23-s − 2.70·26-s + 1.28·28-s − 1.24·29-s − 1.21·31-s − 4.22·32-s + 1.58·34-s − 0.391·37-s + 1.64·38-s + 0.262·41-s + 0.0367·43-s + 1.85·44-s + 0.195·46-s + 1.18·47-s − 0.785·49-s + 3.86·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(70.2866\)
Root analytic conductor: \(8.38371\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 225,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 \)
good2 \( 1 + 11 p T + p^{7} T^{2} \)
7 \( 1 - 60 p T + p^{7} T^{2} \)
11 \( 1 - 2944 T + p^{7} T^{2} \)
13 \( 1 - 11006 T + p^{7} T^{2} \)
17 \( 1 + 16546 T + p^{7} T^{2} \)
19 \( 1 + 25364 T + p^{7} T^{2} \)
23 \( 1 + 5880 T + p^{7} T^{2} \)
29 \( 1 + 163042 T + p^{7} T^{2} \)
31 \( 1 + 201600 T + p^{7} T^{2} \)
37 \( 1 + 120530 T + p^{7} T^{2} \)
41 \( 1 - 115910 T + p^{7} T^{2} \)
43 \( 1 - 19148 T + p^{7} T^{2} \)
47 \( 1 - 841016 T + p^{7} T^{2} \)
53 \( 1 - 501890 T + p^{7} T^{2} \)
59 \( 1 - 1586176 T + p^{7} T^{2} \)
61 \( 1 + 372962 T + p^{7} T^{2} \)
67 \( 1 + 4561044 T + p^{7} T^{2} \)
71 \( 1 + 1512832 T + p^{7} T^{2} \)
73 \( 1 - 1522910 T + p^{7} T^{2} \)
79 \( 1 - 4231920 T + p^{7} T^{2} \)
83 \( 1 + 1854204 T + p^{7} T^{2} \)
89 \( 1 - 6888174 T + p^{7} T^{2} \)
97 \( 1 + 3700034 T + p^{7} T^{2} \)
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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

−10.53413088965583313856134934619, −9.163567490664873772247012618462, −8.790403806774022468330852539037, −7.77665382490528321309050681873, −6.77031478840877125404712810918, −5.88405856826391183212018429011, −3.75263034519372789688621750143, −2.13079495993701829665779341520, −1.27802189869697203127796031207, 0, 1.27802189869697203127796031207, 2.13079495993701829665779341520, 3.75263034519372789688621750143, 5.88405856826391183212018429011, 6.77031478840877125404712810918, 7.77665382490528321309050681873, 8.790403806774022468330852539037, 9.163567490664873772247012618462, 10.53413088965583313856134934619

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