Properties

Label 2-15e2-1.1-c5-0-7
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $36.0863$
Root an. cond. $6.00719$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 28·4-s + 132·7-s + 120·8-s − 472·11-s + 686·13-s − 264·14-s + 656·16-s − 1.56e3·17-s − 2.18e3·19-s + 944·22-s + 264·23-s − 1.37e3·26-s − 3.69e3·28-s − 170·29-s + 7.27e3·31-s − 5.15e3·32-s + 3.12e3·34-s + 142·37-s + 4.36e3·38-s + 1.61e4·41-s + 1.03e4·43-s + 1.32e4·44-s − 528·46-s + 1.85e4·47-s + 617·49-s − 1.92e4·52-s + ⋯
L(s)  = 1  − 0.353·2-s − 7/8·4-s + 1.01·7-s + 0.662·8-s − 1.17·11-s + 1.12·13-s − 0.359·14-s + 0.640·16-s − 1.31·17-s − 1.38·19-s + 0.415·22-s + 0.104·23-s − 0.398·26-s − 0.890·28-s − 0.0375·29-s + 1.35·31-s − 0.889·32-s + 0.463·34-s + 0.0170·37-s + 0.489·38-s + 1.50·41-s + 0.850·43-s + 1.02·44-s − 0.0367·46-s + 1.22·47-s + 0.0367·49-s − 0.985·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(36.0863\)
Root analytic conductor: \(6.00719\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.257405093\)
\(L(\frac12)\) \(\approx\) \(1.257405093\)
\(L(\frac{7}{2})\) 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 \( 1 + p T + p^{5} T^{2} \)
7 \( 1 - 132 T + p^{5} T^{2} \)
11 \( 1 + 472 T + p^{5} T^{2} \)
13 \( 1 - 686 T + p^{5} T^{2} \)
17 \( 1 + 1562 T + p^{5} T^{2} \)
19 \( 1 + 2180 T + p^{5} T^{2} \)
23 \( 1 - 264 T + p^{5} T^{2} \)
29 \( 1 + 170 T + p^{5} T^{2} \)
31 \( 1 - 7272 T + p^{5} T^{2} \)
37 \( 1 - 142 T + p^{5} T^{2} \)
41 \( 1 - 16198 T + p^{5} T^{2} \)
43 \( 1 - 10316 T + p^{5} T^{2} \)
47 \( 1 - 18568 T + p^{5} T^{2} \)
53 \( 1 - 21514 T + p^{5} T^{2} \)
59 \( 1 + 34600 T + p^{5} T^{2} \)
61 \( 1 + 35738 T + p^{5} T^{2} \)
67 \( 1 - 5772 T + p^{5} T^{2} \)
71 \( 1 - 69088 T + p^{5} T^{2} \)
73 \( 1 - 70526 T + p^{5} T^{2} \)
79 \( 1 - 47640 T + p^{5} T^{2} \)
83 \( 1 - 74004 T + p^{5} T^{2} \)
89 \( 1 - 90030 T + p^{5} T^{2} \)
97 \( 1 - 33502 T + p^{5} T^{2} \)
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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

−10.87990883679588545063734019119, −10.71019666781469943032338107378, −9.203573229402758379945278982473, −8.419337515910641690074033186641, −7.77447518440058863391877949415, −6.19219048924158129641597311644, −4.90851025181864403280587021437, −4.12331881089028883278302644378, −2.23988110029223652967370459876, −0.72031908795903763908152315873, 0.72031908795903763908152315873, 2.23988110029223652967370459876, 4.12331881089028883278302644378, 4.90851025181864403280587021437, 6.19219048924158129641597311644, 7.77447518440058863391877949415, 8.419337515910641690074033186641, 9.203573229402758379945278982473, 10.71019666781469943032338107378, 10.87990883679588545063734019119

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