Properties

Label 2-15e2-5.4-c1-0-4
Degree $2$
Conductor $225$
Sign $0.894 + 0.447i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 5i·7-s + 5i·13-s + 4·16-s + 19-s − 10i·28-s − 7·31-s + 10i·37-s + 5i·43-s − 18·49-s + 10i·52-s − 13·61-s + 8·64-s − 5i·67-s − 10i·73-s + ⋯
L(s)  = 1  + 4-s − 1.88i·7-s + 1.38i·13-s + 16-s + 0.229·19-s − 1.88i·28-s − 1.25·31-s + 1.64i·37-s + 0.762i·43-s − 2.57·49-s + 1.38i·52-s − 1.66·61-s + 64-s − 0.610i·67-s − 1.17i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44034 - 0.340019i\)
\(L(\frac12)\) \(\approx\) \(1.44034 - 0.340019i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 2T^{2} \)
7 \( 1 + 5iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 5iT - 97T^{2} \)
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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

−11.97213337598336306677428842685, −11.13023079679541338647056538270, −10.45407558964146273602629526461, −9.446957461320592505642302542068, −7.87110470045488970311903019173, −7.09361384705354003603112536606, −6.36362376933091395923522437303, −4.59633600622430725213246465689, −3.42938708516637227639041179406, −1.55320214804596082455230921611, 2.15579548274090720163004865639, 3.20639718690438586361714665153, 5.42886869452132221315686201870, 5.92646873641032548968684219242, 7.32422040193434457526132742217, 8.348809616381345941528354267912, 9.342256799032793945735448481535, 10.55139120736728572680468902968, 11.42765041955848287325130166705, 12.34436268877773826824390940586

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