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Properties

Label 2-15e2-25.6-c1-0-3
Degree $2$
Conductor $225$
Sign $0.299 - 0.954i$
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  + (0.346 + 1.06i)2-s + (0.598 − 0.435i)4-s + (−0.407 + 2.19i)5-s + 1.11·7-s + (2.48 + 1.80i)8-s + (−2.48 + 0.327i)10-s + (−1.13 − 3.49i)11-s + (−1.25 + 3.85i)13-s + (0.386 + 1.18i)14-s + (−0.609 + 1.87i)16-s + (1.71 + 1.24i)17-s + (3.28 + 2.38i)19-s + (0.712 + 1.49i)20-s + (3.33 − 2.42i)22-s + (−1.90 − 5.87i)23-s + ⋯
L(s)  = 1  + (0.245 + 0.754i)2-s + (0.299 − 0.217i)4-s + (−0.182 + 0.983i)5-s + 0.420·7-s + (0.879 + 0.639i)8-s + (−0.786 + 0.103i)10-s + (−0.341 − 1.05i)11-s + (−0.347 + 1.06i)13-s + (0.103 + 0.317i)14-s + (−0.152 + 0.468i)16-s + (0.416 + 0.302i)17-s + (0.753 + 0.547i)19-s + (0.159 + 0.334i)20-s + (0.710 − 0.516i)22-s + (−0.397 − 1.22i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.299 - 0.954i$
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: \((2,\ 225,\ (\ :1/2),\ 0.299 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25351 + 0.920365i\)
\(L(\frac12)\) \(\approx\) \(1.25351 + 0.920365i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.407 - 2.19i)T \)
good2 \( 1 + (-0.346 - 1.06i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 - 1.11T + 7T^{2} \)
11 \( 1 + (1.13 + 3.49i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.25 - 3.85i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.71 - 1.24i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.28 - 2.38i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.90 + 5.87i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (1.82 - 1.32i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (8.13 + 5.90i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.27 + 7.01i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.30 + 7.10i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.24T + 43T^{2} \)
47 \( 1 + (2.53 - 1.83i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.83 - 2.06i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.03 + 6.27i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.81 + 8.65i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (2.12 + 1.54i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-0.534 + 0.388i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.31 - 7.10i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.90 - 5.01i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.92 - 7.20i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.72 - 14.5i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (10.9 - 7.93i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.39309025005559033098814905291, −11.10722275628168173158103703465, −10.91294563039182746033552720159, −9.544552330458933773301981348726, −8.064733580075198978564057781059, −7.36075577392080630399209854890, −6.30756809958005071283931267947, −5.47802146853100532168494558768, −3.92087209119067758734920433397, −2.23824610712082193020406775348, 1.52833873265772016646555835481, 3.10943904590857521620809716270, 4.52037347494811286217555933965, 5.43597102201087469605881456947, 7.38156104990628656244342414304, 7.86165889575085899861173507339, 9.366363306929242370229466027372, 10.19989050092674723831145955108, 11.35451274948627412371958437286, 12.06514071283581820550062322136

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