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Properties

Label 2-15e2-25.6-c1-0-1
Degree $2$
Conductor $225$
Sign $-0.996 - 0.0880i$
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  + (0.670 + 2.06i)2-s + (−2.18 + 1.59i)4-s + (−1.69 + 1.45i)5-s − 3.32·7-s + (−1.23 − 0.900i)8-s + (−4.14 − 2.52i)10-s + (1.18 + 3.63i)11-s + (0.819 − 2.52i)13-s + (−2.22 − 6.85i)14-s + (−0.645 + 1.98i)16-s + (3.82 + 2.78i)17-s + (2.95 + 2.14i)19-s + (1.39 − 5.88i)20-s + (−6.70 + 4.87i)22-s + (−1.11 − 3.44i)23-s + ⋯
L(s)  = 1  + (0.474 + 1.45i)2-s + (−1.09 + 0.795i)4-s + (−0.758 + 0.651i)5-s − 1.25·7-s + (−0.438 − 0.318i)8-s + (−1.31 − 0.797i)10-s + (0.355 + 1.09i)11-s + (0.227 − 0.699i)13-s + (−0.595 − 1.83i)14-s + (−0.161 + 0.496i)16-s + (0.928 + 0.674i)17-s + (0.677 + 0.492i)19-s + (0.311 − 1.31i)20-s + (−1.42 + 1.03i)22-s + (−0.233 − 0.717i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.996 - 0.0880i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.996 - 0.0880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0508570 + 1.15320i\)
\(L(\frac12)\) \(\approx\) \(0.0508570 + 1.15320i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 1.45i)T \)
good2 \( 1 + (-0.670 - 2.06i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + 3.32T + 7T^{2} \)
11 \( 1 + (-1.18 - 3.63i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.819 + 2.52i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.82 - 2.78i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.95 - 2.14i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.11 + 3.44i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.0281 + 0.0204i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-8.17 - 5.94i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.23 + 9.95i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.589 - 1.81i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.27T + 43T^{2} \)
47 \( 1 + (-1.19 + 0.867i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (7.22 - 5.24i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.83 - 11.7i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.59 + 4.92i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-4.39 - 3.19i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-9.80 + 7.12i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.78 - 11.6i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.86 + 4.26i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.55 + 3.30i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (5.50 + 16.9i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-6.69 + 4.86i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.72445358340026386494638307709, −12.22119431422960709562605781409, −10.61977042757346564534968758965, −9.746774755966009356447742215746, −8.304737259550185536672545910331, −7.46082453316137267656498484105, −6.65133177579223031224533756011, −5.83213743040537268096847175264, −4.35624973096713508800221559744, −3.25064314583645675017677242316, 0.887165544824935396953054976513, 3.05329820103243741257811034397, 3.76195780380235561695295066423, 5.04167600963952367333723926619, 6.50508582253497776738190719627, 7.983636505620321880048081813308, 9.349018390820394984274819408902, 9.812110711385840464178817319676, 11.29502786138150088789901395118, 11.70497788701127087263019717470

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