Properties

Label 2-15e2-25.21-c3-0-25
Degree $2$
Conductor $225$
Sign $0.893 - 0.449i$
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  + (0.0492 − 0.151i)2-s + (6.45 + 4.68i)4-s + (9.69 + 5.56i)5-s + 29.3·7-s + (2.06 − 1.49i)8-s + (1.32 − 1.19i)10-s + (10.1 − 31.2i)11-s + (−9.02 − 27.7i)13-s + (1.44 − 4.45i)14-s + (19.5 + 60.2i)16-s + (−103. + 75.3i)17-s + (3.24 − 2.36i)19-s + (36.4 + 81.3i)20-s + (−4.24 − 3.08i)22-s + (39.8 − 122. i)23-s + ⋯
L(s)  = 1  + (0.0174 − 0.0536i)2-s + (0.806 + 0.585i)4-s + (0.867 + 0.497i)5-s + 1.58·7-s + (0.0910 − 0.0661i)8-s + (0.0417 − 0.0378i)10-s + (0.278 − 0.857i)11-s + (−0.192 − 0.592i)13-s + (0.0276 − 0.0849i)14-s + (0.306 + 0.941i)16-s + (−1.47 + 1.07i)17-s + (0.0392 − 0.0284i)19-s + (0.407 + 0.909i)20-s + (−0.0411 − 0.0298i)22-s + (0.361 − 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.893 - 0.449i$
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: \((2,\ 225,\ (\ :3/2),\ 0.893 - 0.449i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.78215 + 0.660565i\)
\(L(\frac12)\) \(\approx\) \(2.78215 + 0.660565i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.69 - 5.56i)T \)
good2 \( 1 + (-0.0492 + 0.151i)T + (-6.47 - 4.70i)T^{2} \)
7 \( 1 - 29.3T + 343T^{2} \)
11 \( 1 + (-10.1 + 31.2i)T + (-1.07e3 - 782. i)T^{2} \)
13 \( 1 + (9.02 + 27.7i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (103. - 75.3i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-3.24 + 2.36i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + (-39.8 + 122. i)T + (-9.84e3 - 7.15e3i)T^{2} \)
29 \( 1 + (178. + 129. i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (-139. + 101. i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-64.2 - 197. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-78.0 - 240. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 + 81.1T + 7.95e4T^{2} \)
47 \( 1 + (281. + 204. i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (324. + 236. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (-196. - 604. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (111. - 341. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + (72.2 - 52.4i)T + (9.29e4 - 2.86e5i)T^{2} \)
71 \( 1 + (-434. - 315. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (-28.1 + 86.7i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (1.07e3 + 782. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (148. - 107. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + (179. - 552. i)T + (-5.70e5 - 4.14e5i)T^{2} \)
97 \( 1 + (1.34e3 + 975. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−11.46095018633801593209926228153, −11.14902537683611837743126520026, −10.22812539376133643064676093115, −8.635295349160125165449713175000, −7.998227591272663908549416763740, −6.71909023478751741574446428108, −5.84080211545912281563936931153, −4.36066682856717485852652841679, −2.73385370780458085811063222562, −1.67323071529636162596525106323, 1.47397179901387886460239700773, 2.20064065017969718506523228915, 4.67347978168883132822069303292, 5.29633245171321528961936365696, 6.67219754102745799549839112705, 7.53843752922712900685598005751, 8.973481925866508022265391937837, 9.701285495512236821889875578158, 11.01109095891323770181146860023, 11.45610295662194218567263564054

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