Properties

Label 2-15e2-225.4-c3-0-49
Degree $2$
Conductor $225$
Sign $-0.995 - 0.0929i$
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  + (−1.13 − 5.35i)2-s + (−4.50 + 2.58i)3-s + (−20.0 + 8.91i)4-s + (2.25 − 10.9i)5-s + (18.9 + 21.1i)6-s + (24.9 − 14.3i)7-s + (44.7 + 61.5i)8-s + (13.6 − 23.2i)9-s + (−61.1 + 0.366i)10-s + (43.6 − 9.28i)11-s + (67.2 − 91.8i)12-s + (2.98 − 14.0i)13-s + (−105. − 117. i)14-s + (18.0 + 55.2i)15-s + (161. − 179. i)16-s + (34.9 + 48.0i)17-s + ⋯
L(s)  = 1  + (−0.402 − 1.89i)2-s + (−0.867 + 0.496i)3-s + (−2.50 + 1.11i)4-s + (0.202 − 0.979i)5-s + (1.28 + 1.44i)6-s + (1.34 − 0.777i)7-s + (1.97 + 2.72i)8-s + (0.506 − 0.862i)9-s + (−1.93 + 0.0115i)10-s + (1.19 − 0.254i)11-s + (1.61 − 2.21i)12-s + (0.0637 − 0.299i)13-s + (−2.01 − 2.23i)14-s + (0.311 + 0.950i)15-s + (2.51 − 2.79i)16-s + (0.498 + 0.685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0929i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.995 - 0.0929i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.995 - 0.0929i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0542955 + 1.16546i\)
\(L(\frac12)\) \(\approx\) \(0.0542955 + 1.16546i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.50 - 2.58i)T \)
5 \( 1 + (-2.25 + 10.9i)T \)
good2 \( 1 + (1.13 + 5.35i)T + (-7.30 + 3.25i)T^{2} \)
7 \( 1 + (-24.9 + 14.3i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-43.6 + 9.28i)T + (1.21e3 - 541. i)T^{2} \)
13 \( 1 + (-2.98 + 14.0i)T + (-2.00e3 - 893. i)T^{2} \)
17 \( 1 + (-34.9 - 48.0i)T + (-1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (-98.4 + 71.5i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + (34.7 - 31.2i)T + (1.27e3 - 1.21e4i)T^{2} \)
29 \( 1 + (12.6 - 120. i)T + (-2.38e4 - 5.07e3i)T^{2} \)
31 \( 1 + (14.3 + 136. i)T + (-2.91e4 + 6.19e3i)T^{2} \)
37 \( 1 + (13.1 - 4.27i)T + (4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (-307. - 65.4i)T + (6.29e4 + 2.80e4i)T^{2} \)
43 \( 1 + (-226. + 130. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (285. + 30.0i)T + (1.01e5 + 2.15e4i)T^{2} \)
53 \( 1 + (-279. + 385. i)T + (-4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (265. + 56.5i)T + (1.87e5 + 8.35e4i)T^{2} \)
61 \( 1 + (1.87 - 0.398i)T + (2.07e5 - 9.23e4i)T^{2} \)
67 \( 1 + (841. - 88.4i)T + (2.94e5 - 6.25e4i)T^{2} \)
71 \( 1 + (-256. - 186. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (272. + 88.5i)T + (3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-7.68 + 73.1i)T + (-4.82e5 - 1.02e5i)T^{2} \)
83 \( 1 + (345. - 775. i)T + (-3.82e5 - 4.24e5i)T^{2} \)
89 \( 1 + (-145. + 448. i)T + (-5.70e5 - 4.14e5i)T^{2} \)
97 \( 1 + (750. + 78.8i)T + (8.92e5 + 1.89e5i)T^{2} \)
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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

−11.39960080458262847085686310466, −10.55124364256809774190253448641, −9.607138105477710831880179030505, −8.877305251335106511062243491622, −7.74127218035283971094324196601, −5.43482748781844028888799150009, −4.51045809582483206574844730875, −3.74417812819172932881534919853, −1.45993360789190226943520435819, −0.815579071824017961881191084621, 1.39231739319639089500721661416, 4.45369505501045750238875549846, 5.58210203946294428017461253787, 6.19652539614310109993029004021, 7.28929250458108753408909607379, 7.81872295285464428952357648050, 9.097146307360280725659342101427, 10.08374979823002922965551118493, 11.36662001262477778798950244508, 12.21943160331690779609400855739

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