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Properties

Label 2-15e2-45.32-c3-0-13
Degree $2$
Conductor $225$
Sign $0.535 - 0.844i$
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  + (−3.56 − 0.956i)2-s + (−4.00 + 3.31i)3-s + (4.89 + 2.82i)4-s + (17.4 − 8.01i)6-s + (−1.11 + 4.17i)7-s + (6.12 + 6.12i)8-s + (5.00 − 26.5i)9-s + (−19.8 + 11.4i)11-s + (−28.9 + 4.93i)12-s + (−10.6 − 39.8i)13-s + (7.98 − 13.8i)14-s + (−38.6 − 66.9i)16-s + (53.3 − 53.3i)17-s + (−43.2 + 89.9i)18-s − 63.9i·19-s + ⋯
L(s)  = 1  + (−1.26 − 0.338i)2-s + (−0.769 + 0.638i)3-s + (0.612 + 0.353i)4-s + (1.18 − 0.545i)6-s + (−0.0604 + 0.225i)7-s + (0.270 + 0.270i)8-s + (0.185 − 0.982i)9-s + (−0.542 + 0.313i)11-s + (−0.697 + 0.118i)12-s + (−0.228 − 0.851i)13-s + (0.152 − 0.264i)14-s + (−0.603 − 1.04i)16-s + (0.761 − 0.761i)17-s + (−0.566 + 1.17i)18-s − 0.772i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.535 - 0.844i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.402902 + 0.221543i\)
\(L(\frac12)\) \(\approx\) \(0.402902 + 0.221543i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.00 - 3.31i)T \)
5 \( 1 \)
good2 \( 1 + (3.56 + 0.956i)T + (6.92 + 4i)T^{2} \)
7 \( 1 + (1.11 - 4.17i)T + (-297. - 171.5i)T^{2} \)
11 \( 1 + (19.8 - 11.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (10.6 + 39.8i)T + (-1.90e3 + 1.09e3i)T^{2} \)
17 \( 1 + (-53.3 + 53.3i)T - 4.91e3iT^{2} \)
19 \( 1 + 63.9iT - 6.85e3T^{2} \)
23 \( 1 + (141. - 37.9i)T + (1.05e4 - 6.08e3i)T^{2} \)
29 \( 1 + (-145. - 252. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-60.4 + 104. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-132. - 132. i)T + 5.06e4iT^{2} \)
41 \( 1 + (-169. - 97.9i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (183. + 49.1i)T + (6.88e4 + 3.97e4i)T^{2} \)
47 \( 1 + (277. + 74.4i)T + (8.99e4 + 5.19e4i)T^{2} \)
53 \( 1 + (-508. - 508. i)T + 1.48e5iT^{2} \)
59 \( 1 + (344. - 597. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (98.3 + 170. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-684. + 183. i)T + (2.60e5 - 1.50e5i)T^{2} \)
71 \( 1 + 192. iT - 3.57e5T^{2} \)
73 \( 1 + (257. - 257. i)T - 3.89e5iT^{2} \)
79 \( 1 + (543. - 313. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (105. - 395. i)T + (-4.95e5 - 2.85e5i)T^{2} \)
89 \( 1 - 231.T + 7.04e5T^{2} \)
97 \( 1 + (139. - 519. i)T + (-7.90e5 - 4.56e5i)T^{2} \)
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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

−11.65776559587331481004482672070, −10.66325490696156576868576214444, −10.04164919170676862139142343449, −9.332839245589002773803479300796, −8.200520752055807711447469240736, −7.16432167186949280751534617730, −5.64908661175529677017964576010, −4.69685710317637468555545698281, −2.82982472664317823616412397266, −0.874358755335986402393020819740, 0.46050000894117764577848464430, 1.89071942900378038147356303470, 4.23725376787554166426573615209, 5.85103171196282401013361188061, 6.73433031161767255848706846971, 7.84280741358692307088934346221, 8.344014220316706406994298325766, 9.896656193611600336300595444030, 10.35755729567771602487937556572, 11.51899585005035764480642517089

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