Properties

Label 2-15e2-25.21-c3-0-18
Degree $2$
Conductor $225$
Sign $0.922 - 0.386i$
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  + (−1.48 + 4.58i)2-s + (−12.3 − 8.93i)4-s + (−5.89 + 9.49i)5-s + 1.13·7-s + (28.0 − 20.4i)8-s + (−34.7 − 41.1i)10-s + (17.7 − 54.7i)11-s + (−12.9 − 39.9i)13-s + (−1.68 + 5.19i)14-s + (14.0 + 43.3i)16-s + (−87.1 + 63.3i)17-s + (43.1 − 31.3i)19-s + (157. − 64.1i)20-s + (224. + 162. i)22-s + (5.32 − 16.3i)23-s + ⋯
L(s)  = 1  + (−0.526 + 1.61i)2-s + (−1.53 − 1.11i)4-s + (−0.527 + 0.849i)5-s + 0.0612·7-s + (1.24 − 0.901i)8-s + (−1.09 − 1.30i)10-s + (0.487 − 1.49i)11-s + (−0.276 − 0.851i)13-s + (−0.0322 + 0.0992i)14-s + (0.219 + 0.677i)16-s + (−1.24 + 0.903i)17-s + (0.520 − 0.378i)19-s + (1.76 − 0.716i)20-s + (2.17 + 1.57i)22-s + (0.0482 − 0.148i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.922 - 0.386i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.922 - 0.386i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.666689 + 0.133894i\)
\(L(\frac12)\) \(\approx\) \(0.666689 + 0.133894i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.89 - 9.49i)T \)
good2 \( 1 + (1.48 - 4.58i)T + (-6.47 - 4.70i)T^{2} \)
7 \( 1 - 1.13T + 343T^{2} \)
11 \( 1 + (-17.7 + 54.7i)T + (-1.07e3 - 782. i)T^{2} \)
13 \( 1 + (12.9 + 39.9i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (87.1 - 63.3i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-43.1 + 31.3i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + (-5.32 + 16.3i)T + (-9.84e3 - 7.15e3i)T^{2} \)
29 \( 1 + (-113. - 82.4i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (-196. + 142. i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-107. - 332. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-2.64 - 8.13i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 - 111.T + 7.95e4T^{2} \)
47 \( 1 + (261. + 189. i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (332. + 241. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (205. + 633. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (-91.5 + 281. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + (-679. + 494. i)T + (9.29e4 - 2.86e5i)T^{2} \)
71 \( 1 + (45.5 + 33.1i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (-66.9 + 206. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (339. + 246. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-1.17e3 + 856. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + (-378. + 1.16e3i)T + (-5.70e5 - 4.14e5i)T^{2} \)
97 \( 1 + (419. + 305. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−11.62240530506129594823698712057, −10.76557512487320113835511548184, −9.629839467740796386336766275815, −8.364351477928532013276965886548, −8.018553339102727446340160195596, −6.62116416010225263500395972594, −6.23150930084054400965695258765, −4.79089338421846586230451805775, −3.20017785781522538716518202529, −0.38741010214262435573950824241, 1.21479504714986733763069362344, 2.44163088480707814780508163815, 4.13651079308776632357913018272, 4.72333759710539541638788199316, 6.93174740840810312600034868720, 8.166575186555890847368574766455, 9.336534976092893209630051862813, 9.575722785332930872699988840107, 10.93345513481539036699134402556, 11.87030617131974596872275069171

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