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Properties

Label 2-15e2-9.2-c2-0-9
Degree $2$
Conductor $225$
Sign $-0.837 - 0.545i$
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  + (2.46 + 1.42i)2-s + (−0.600 + 2.93i)3-s + (2.03 + 3.52i)4-s + (−5.65 + 6.38i)6-s + (−4.86 + 8.42i)7-s + 0.212i·8-s + (−8.27 − 3.52i)9-s + (0.370 + 0.214i)11-s + (−11.5 + 3.87i)12-s + (9.37 + 16.2i)13-s + (−23.9 + 13.8i)14-s + (7.84 − 13.5i)16-s + 2.85i·17-s + (−15.3 − 20.4i)18-s − 0.530·19-s + ⋯
L(s)  = 1  + (1.23 + 0.710i)2-s + (−0.200 + 0.979i)3-s + (0.509 + 0.882i)4-s + (−0.942 + 1.06i)6-s + (−0.694 + 1.20i)7-s + 0.0265i·8-s + (−0.919 − 0.392i)9-s + (0.0337 + 0.0194i)11-s + (−0.966 + 0.322i)12-s + (0.721 + 1.24i)13-s + (−1.71 + 0.987i)14-s + (0.490 − 0.849i)16-s + 0.168i·17-s + (−0.853 − 1.13i)18-s − 0.0279·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.837 - 0.545i$
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: \((2,\ 225,\ (\ :1),\ -0.837 - 0.545i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.686743 + 2.31209i\)
\(L(\frac12)\) \(\approx\) \(0.686743 + 2.31209i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.600 - 2.93i)T \)
5 \( 1 \)
good2 \( 1 + (-2.46 - 1.42i)T + (2 + 3.46i)T^{2} \)
7 \( 1 + (4.86 - 8.42i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-0.370 - 0.214i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-9.37 - 16.2i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 2.85iT - 289T^{2} \)
19 \( 1 + 0.530T + 361T^{2} \)
23 \( 1 + (-18.8 + 10.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-21.0 - 12.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-6.33 - 10.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 14.5T + 1.36e3T^{2} \)
41 \( 1 + (33.1 - 19.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-28.8 + 50.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-42.9 - 24.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 44.5iT - 2.80e3T^{2} \)
59 \( 1 + (-54.6 + 31.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-11.0 + 19.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-16.2 - 28.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 89.8iT - 5.04e3T^{2} \)
73 \( 1 + 144.T + 5.32e3T^{2} \)
79 \( 1 + (-25.1 + 43.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-66.2 - 38.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 28.9iT - 7.92e3T^{2} \)
97 \( 1 + (-11.4 + 19.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.41150277813716372882671114181, −11.79584498613872897058140403557, −10.55299446714923884301714136764, −9.333918067721145249917756691318, −8.677111374028770299325413802765, −6.77658485140467229577733710764, −6.05547535170634672722649903939, −5.12454450045796309485909957647, −4.08911336951750427498919744179, −2.95928366315594678021589372626, 0.974625197940877947043555443232, 2.80084204855406240385954710676, 3.80513170237358157461732893238, 5.26792226417593058346771158615, 6.28115383924394621176109600435, 7.35402869664980973507701899613, 8.460198562080837775774654723488, 10.20730336260734342435188675333, 10.96277015323481540726216383743, 11.85369611371671923684578111022

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