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Properties

Label 2-15e2-45.4-c1-0-4
Degree $2$
Conductor $225$
Sign $0.801 - 0.598i$
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  + (0.409 − 0.236i)2-s + (−1.64 − 0.544i)3-s + (−0.888 + 1.53i)4-s + (−0.802 + 0.165i)6-s + (2.21 − 1.28i)7-s + 1.78i·8-s + (2.40 + 1.79i)9-s + (3.08 + 5.34i)11-s + (2.29 − 2.04i)12-s + (1.84 + 1.06i)13-s + (0.606 − 1.05i)14-s + (−1.35 − 2.34i)16-s + 3.16i·17-s + (1.41 + 0.164i)18-s − 0.356·19-s + ⋯
L(s)  = 1  + (0.289 − 0.167i)2-s + (−0.949 − 0.314i)3-s + (−0.444 + 0.769i)4-s + (−0.327 + 0.0676i)6-s + (0.838 − 0.484i)7-s + 0.631i·8-s + (0.802 + 0.597i)9-s + (0.929 + 1.61i)11-s + (0.663 − 0.590i)12-s + (0.512 + 0.295i)13-s + (0.162 − 0.280i)14-s + (−0.338 − 0.585i)16-s + 0.768i·17-s + (0.332 + 0.0388i)18-s − 0.0817·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.801 - 0.598i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.801 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00586 + 0.334235i\)
\(L(\frac12)\) \(\approx\) \(1.00586 + 0.334235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.64 + 0.544i)T \)
5 \( 1 \)
good2 \( 1 + (-0.409 + 0.236i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-2.21 + 1.28i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.08 - 5.34i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.84 - 1.06i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.16iT - 17T^{2} \)
19 \( 1 + 0.356T + 19T^{2} \)
23 \( 1 + (3.64 + 2.10i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.843 - 1.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.12 + 7.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.63iT - 37T^{2} \)
41 \( 1 + (-1.36 + 2.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.64 + 3.83i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.89 - 5.71i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.43iT - 53T^{2} \)
59 \( 1 + (-5.10 + 8.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.00549 - 0.00952i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.851 + 0.491i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.43T + 71T^{2} \)
73 \( 1 + 6.61iT - 73T^{2} \)
79 \( 1 + (4.73 + 8.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.02 + 5.20i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.26T + 89T^{2} \)
97 \( 1 + (-6.24 + 3.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.23915292463853026659013087426, −11.65820403144688877595994207860, −10.67474326842067744927715170515, −9.530294729635153848940987124530, −8.162652218516779087108491720874, −7.33365563178183307263120935483, −6.20581194842514725298063794735, −4.61855018126862700499861405801, −4.17879993150475708284605420334, −1.79164873623545839693401092212, 1.07090917702538294802670086024, 3.77999135673285142353595879688, 5.01478998702267543128722573935, 5.79623822581229588180701659450, 6.59803486229813285368226675192, 8.362975068292757906321068581335, 9.287782574998780209527694013102, 10.37084834523574826270617435529, 11.30881272597648939994245433633, 11.83149190909191853417970194202

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