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Properties

Label 2-15e2-9.2-c2-0-8
Degree $2$
Conductor $225$
Sign $0.676 + 0.736i$
Analytic cond. $6.13080$
Root an. cond. $2.47604$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.31 − 1.91i)2-s + (0.962 − 2.84i)3-s + (5.32 + 9.21i)4-s + (−8.62 + 7.57i)6-s + (−2.98 + 5.17i)7-s − 25.4i·8-s + (−7.14 − 5.47i)9-s + (8.27 + 4.77i)11-s + (31.3 − 6.24i)12-s + (7.79 + 13.5i)13-s + (19.8 − 11.4i)14-s + (−27.3 + 47.3i)16-s + 13.6i·17-s + (13.2 + 31.8i)18-s + 19.9·19-s + ⋯
L(s)  = 1  + (−1.65 − 0.956i)2-s + (0.320 − 0.947i)3-s + (1.33 + 2.30i)4-s + (−1.43 + 1.26i)6-s + (−0.426 + 0.739i)7-s − 3.17i·8-s + (−0.794 − 0.607i)9-s + (0.752 + 0.434i)11-s + (2.60 − 0.520i)12-s + (0.599 + 1.03i)13-s + (1.41 − 0.816i)14-s + (−1.70 + 2.96i)16-s + 0.804i·17-s + (0.734 + 1.76i)18-s + 1.04·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(6.13080\)
Root analytic conductor: \(2.47604\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1),\ 0.676 + 0.736i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.693594 - 0.304704i\)
\(L(\frac12)\) \(\approx\) \(0.693594 - 0.304704i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.962 + 2.84i)T \)
5 \( 1 \)
good2 \( 1 + (3.31 + 1.91i)T + (2 + 3.46i)T^{2} \)
7 \( 1 + (2.98 - 5.17i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-8.27 - 4.77i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.79 - 13.5i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 13.6iT - 289T^{2} \)
19 \( 1 - 19.9T + 361T^{2} \)
23 \( 1 + (-20.9 + 12.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-6.28 - 3.63i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (2.80 + 4.86i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 23.4T + 1.36e3T^{2} \)
41 \( 1 + (-13.6 + 7.88i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-10.2 + 17.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (64.0 + 37.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 23.3iT - 2.80e3T^{2} \)
59 \( 1 + (28.3 - 16.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-26.9 + 46.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-63.3 - 109. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 14.9iT - 5.04e3T^{2} \)
73 \( 1 - 45.7T + 5.32e3T^{2} \)
79 \( 1 + (-12.0 + 20.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-78.3 - 45.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 51.6iT - 7.92e3T^{2} \)
97 \( 1 + (-71.9 + 124. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.80980486916889613900411540664, −11.01005702423162348421238057237, −9.607333897971637041485851992470, −9.041371254932547949280287342089, −8.286141833605398689514270282867, −7.13901792840101857476894143216, −6.35504756364682541417575651741, −3.58353330672993442330867993113, −2.33347496332014209553842861334, −1.20179149022204895739074462324, 0.837762428888446811740199646628, 3.22855169306907705823276442647, 5.15534162524692390712923679153, 6.26952952629924728315121233681, 7.43055641436433597249497837895, 8.282470019818553461651121134364, 9.327042206477063722411438105819, 9.776711484209230928455645709525, 10.79567556601356419498830463963, 11.40455424671383693002751013742

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