Properties

Label 2-15e2-225.16-c1-0-14
Degree $2$
Conductor $225$
Sign $0.994 - 0.108i$
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  + (−2.06 + 0.917i)2-s + (1.65 + 0.510i)3-s + (2.06 − 2.29i)4-s + (−2.23 − 0.160i)5-s + (−3.87 + 0.465i)6-s + (2.12 − 3.67i)7-s + (−0.759 + 2.33i)8-s + (2.47 + 1.69i)9-s + (4.74 − 1.71i)10-s + (2.79 − 1.24i)11-s + (4.59 − 2.74i)12-s + (−2.17 − 0.969i)13-s + (−1.00 + 9.53i)14-s + (−3.60 − 1.40i)15-s + (0.0664 + 0.632i)16-s + (2.36 − 7.29i)17-s + ⋯
L(s)  = 1  + (−1.45 + 0.648i)2-s + (0.955 + 0.294i)3-s + (1.03 − 1.14i)4-s + (−0.997 − 0.0718i)5-s + (−1.58 + 0.190i)6-s + (0.802 − 1.39i)7-s + (−0.268 + 0.826i)8-s + (0.825 + 0.563i)9-s + (1.50 − 0.542i)10-s + (0.844 − 0.375i)11-s + (1.32 − 0.791i)12-s + (−0.604 − 0.268i)13-s + (−0.267 + 2.54i)14-s + (−0.931 − 0.362i)15-s + (0.0166 + 0.158i)16-s + (0.574 − 1.76i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.994 - 0.108i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.994 - 0.108i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.787864 + 0.0429995i\)
\(L(\frac12)\) \(\approx\) \(0.787864 + 0.0429995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 - 0.510i)T \)
5 \( 1 + (2.23 + 0.160i)T \)
good2 \( 1 + (2.06 - 0.917i)T + (1.33 - 1.48i)T^{2} \)
7 \( 1 + (-2.12 + 3.67i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.79 + 1.24i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (2.17 + 0.969i)T + (8.69 + 9.66i)T^{2} \)
17 \( 1 + (-2.36 + 7.29i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.781 - 2.40i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.656 - 6.24i)T + (-22.4 - 4.78i)T^{2} \)
29 \( 1 + (-4.40 - 0.935i)T + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (-1.42 + 0.302i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (-0.524 + 0.380i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.84 - 0.823i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (-0.433 + 0.750i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.74 + 2.07i)T + (42.9 + 19.1i)T^{2} \)
53 \( 1 + (0.161 + 0.497i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.08 - 0.482i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (9.93 - 4.42i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (-2.36 + 0.502i)T + (61.2 - 27.2i)T^{2} \)
71 \( 1 + (3.08 + 9.50i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.70 + 1.96i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.44 - 0.307i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (-4.79 - 5.32i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 + (-0.415 - 0.302i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.02 + 0.430i)T + (88.6 + 39.4i)T^{2} \)
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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

−11.90778561527783421317221163318, −10.92970845798493300917422843667, −9.997260519961436899105597478526, −9.208026248099385863835587982527, −8.100127718564225189760166867944, −7.61942546364869331069504793341, −6.97151182208469057252799241518, −4.72432357407131534707461446929, −3.52051132764382526713152330399, −1.11461918547475129617086181720, 1.67223791425643974741524732918, 2.81928562342166203963838343886, 4.43044640946896243861303013707, 6.63835064264265760530549745369, 7.907506035801634932169355187178, 8.414445514364822812971508579750, 9.030871552574998401807260854753, 10.10080764291593291086735881009, 11.22646861749049022752186369946, 12.19020282666558977840533618103

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