Properties

Label 2-15e2-15.8-c1-0-5
Degree $2$
Conductor $225$
Sign $-0.391 + 0.920i$
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  + (−0.707 − 0.707i)2-s − 0.999i·4-s + (2 − 2i)7-s + (−2.12 + 2.12i)8-s − 2.82i·11-s + (−1 − i)13-s − 2.82·14-s + 1.00·16-s + (−2.82 − 2.82i)17-s + (−2.00 + 2.00i)22-s + (2.82 − 2.82i)23-s + 1.41i·26-s + (−1.99 − 1.99i)28-s + 4.24·29-s − 4·31-s + (3.53 + 3.53i)32-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s − 0.499i·4-s + (0.755 − 0.755i)7-s + (−0.750 + 0.750i)8-s − 0.852i·11-s + (−0.277 − 0.277i)13-s − 0.755·14-s + 0.250·16-s + (−0.685 − 0.685i)17-s + (−0.426 + 0.426i)22-s + (0.589 − 0.589i)23-s + 0.277i·26-s + (−0.377 − 0.377i)28-s + 0.787·29-s − 0.718·31-s + (0.624 + 0.624i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.391 + 0.920i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.391 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.502559 - 0.759545i\)
\(L(\frac12)\) \(\approx\) \(0.502559 - 0.759545i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T + 2iT^{2} \)
7 \( 1 + (-2 + 2i)T - 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 + (2.82 + 2.82i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + (-8 - 8i)T + 43iT^{2} \)
47 \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \)
53 \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (4 - 4i)T - 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 - 12iT - 79T^{2} \)
83 \( 1 + (-2.82 + 2.82i)T - 83iT^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + (-11 + 11i)T - 97iT^{2} \)
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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

−11.46959344371491948737042516898, −11.00891156892627751983405687857, −10.16410884146628574775607099986, −9.087039964230170588001187478423, −8.207357606127557863819765877889, −6.95251237299363432805196377715, −5.63791021885016944402708361816, −4.49273148215298350663438184775, −2.67939316892579796084197521046, −0.941842629029834829276222122057, 2.25848312992364474183984224563, 4.01941594741982611705807245109, 5.36123438529372113779581079301, 6.74815821322440349680070626144, 7.60199167602809397153443426663, 8.646272972011338520076968077525, 9.251053975480417444355696257082, 10.54996192158307526478698564559, 11.77595465101173436169065162725, 12.37564912344075466035146942045

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