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Properties

Label 2-15e2-25.21-c1-0-7
Degree $2$
Conductor $225$
Sign $0.638 + 0.769i$
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  + (−0.754 + 2.32i)2-s + (−3.19 − 2.32i)4-s + (−0.824 − 2.07i)5-s − 3.44·7-s + (3.85 − 2.80i)8-s + (5.44 − 0.345i)10-s + (1.00 − 3.10i)11-s + (−0.998 − 3.07i)13-s + (2.59 − 7.98i)14-s + (1.15 + 3.54i)16-s + (−4.08 + 2.97i)17-s + (2.49 − 1.81i)19-s + (−2.19 + 8.56i)20-s + (6.44 + 4.68i)22-s + (−0.478 + 1.47i)23-s + ⋯
L(s)  = 1  + (−0.533 + 1.64i)2-s + (−1.59 − 1.16i)4-s + (−0.368 − 0.929i)5-s − 1.30·7-s + (1.36 − 0.991i)8-s + (1.72 − 0.109i)10-s + (0.304 − 0.936i)11-s + (−0.277 − 0.852i)13-s + (0.693 − 2.13i)14-s + (0.288 + 0.887i)16-s + (−0.991 + 0.720i)17-s + (0.571 − 0.415i)19-s + (−0.490 + 1.91i)20-s + (1.37 + 0.998i)22-s + (−0.0997 + 0.306i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.638 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282133 - 0.132577i\)
\(L(\frac12)\) \(\approx\) \(0.282133 - 0.132577i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.824 + 2.07i)T \)
good2 \( 1 + (0.754 - 2.32i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 + (-1.00 + 3.10i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.998 + 3.07i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.08 - 2.97i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.49 + 1.81i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.478 - 1.47i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.52 + 1.83i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (6.02 - 4.37i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.77 + 5.47i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.67 + 5.15i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.53T + 43T^{2} \)
47 \( 1 + (-5.72 - 4.15i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (8.21 + 5.96i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.534 + 1.64i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.42 + 7.45i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.49 + 1.08i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.577 + 0.419i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.581 - 1.78i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.7 - 7.83i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.20 + 2.32i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.63 - 8.11i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-8.61 - 6.26i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.48055494143868150841240563256, −10.93587702089973318755190222019, −9.550790614865744185403114459775, −8.997569396270392678250622990240, −8.113199819623064939948973251973, −7.09061933476232774852398220734, −6.06121851007196541477271061608, −5.23977877401025412522452295726, −3.67998841499595474887774941723, −0.29680581732251432449195411153, 2.20500720917927950878792245225, 3.30386167669944678712746995166, 4.32610951708912238463306348076, 6.51142388035496001324755967890, 7.44178884976322583304376128652, 9.091507100483169055168251035630, 9.649997467662088778753557530101, 10.44095297188869578458059243220, 11.43999349616241485513258612454, 12.06663726600112934295777310311

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