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Properties

Label 2-15e2-5.4-c3-0-15
Degree $2$
Conductor $225$
Sign $0.447 + 0.894i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·4-s − 20i·7-s − 70i·13-s + 64·16-s − 56·19-s − 160i·28-s + 308·31-s − 110i·37-s − 520i·43-s − 57·49-s − 560i·52-s + 182·61-s + 512·64-s + 880i·67-s + 1.19e3i·73-s + ⋯
L(s)  = 1  + 4-s − 1.07i·7-s − 1.49i·13-s + 16-s − 0.676·19-s − 1.07i·28-s + 1.78·31-s − 0.488i·37-s − 1.84i·43-s − 0.166·49-s − 1.49i·52-s + 0.382·61-s + 64-s + 1.60i·67-s + 1.90i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.80270 - 1.11412i\)
\(L(\frac12)\) \(\approx\) \(1.80270 - 1.11412i\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 - 8T^{2} \)
7 \( 1 + 20iT - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 + 70iT - 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 + 56T + 6.85e3T^{2} \)
23 \( 1 - 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 - 308T + 2.97e4T^{2} \)
37 \( 1 + 110iT - 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 + 520iT - 7.95e4T^{2} \)
47 \( 1 - 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 182T + 2.26e5T^{2} \)
67 \( 1 - 880iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 1.19e3iT - 3.89e5T^{2} \)
79 \( 1 + 884T + 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 1.33e3iT - 9.12e5T^{2} \)
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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

−11.54220132617010680369072812269, −10.48643982641229224489160539567, −10.18413032330575647567067529043, −8.418711145578016408013338854172, −7.52623313079840970827005365401, −6.64584941265074391384830175252, −5.49000046502699681644352355077, −3.91851140829057833404866719228, −2.62208392113583812603295536198, −0.887581425520046305461421266470, 1.77300373682735636680133638761, 2.87111924768286360602338227764, 4.57589828502556953704423015645, 6.05506834780535245079356182681, 6.69011148840149696759225328526, 8.001600599237077787702217421772, 9.008535564018519969191632595058, 10.05937957131318552609303742528, 11.27106770442625783710335372703, 11.83317629619339750275411306096

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