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Properties

Label 2-15e2-45.2-c1-0-5
Degree $2$
Conductor $225$
Sign $-0.599 - 0.800i$
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  + (0.601 + 2.24i)2-s + (1.72 + 0.173i)3-s + (−2.93 + 1.69i)4-s + (0.647 + 3.97i)6-s + (−0.751 + 0.201i)7-s + (−2.29 − 2.29i)8-s + (2.93 + 0.597i)9-s + (−0.220 − 0.127i)11-s + (−5.36 + 2.41i)12-s + (−3.70 − 0.992i)13-s + (−0.903 − 1.56i)14-s + (0.367 − 0.636i)16-s + (3.93 − 3.93i)17-s + (0.427 + 6.95i)18-s + 0.440i·19-s + ⋯
L(s)  = 1  + (0.425 + 1.58i)2-s + (0.994 + 0.100i)3-s + (−1.46 + 0.848i)4-s + (0.264 + 1.62i)6-s + (−0.284 + 0.0761i)7-s + (−0.809 − 0.809i)8-s + (0.979 + 0.199i)9-s + (−0.0663 − 0.0383i)11-s + (−1.54 + 0.697i)12-s + (−1.02 − 0.275i)13-s + (−0.241 − 0.418i)14-s + (0.0918 − 0.159i)16-s + (0.953 − 0.953i)17-s + (0.100 + 1.63i)18-s + 0.101i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 - 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.599 - 0.800i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.599 - 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.827918 + 1.65333i\)
\(L(\frac12)\) \(\approx\) \(0.827918 + 1.65333i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.173i)T \)
5 \( 1 \)
good2 \( 1 + (-0.601 - 2.24i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (0.751 - 0.201i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.220 + 0.127i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.70 + 0.992i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-3.93 + 3.93i)T - 17iT^{2} \)
19 \( 1 - 0.440iT - 19T^{2} \)
23 \( 1 + (-0.917 + 3.42i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.76 + 4.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.0971 + 0.168i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.123 - 0.123i)T + 37iT^{2} \)
41 \( 1 + (3.88 - 2.24i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.357 - 1.33i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.11 + 4.17i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.938 - 0.938i)T + 53iT^{2} \)
59 \( 1 + (-4.02 - 6.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.44 - 2.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.47 - 12.9i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.15iT - 71T^{2} \)
73 \( 1 + (-9.18 + 9.18i)T - 73iT^{2} \)
79 \( 1 + (11.9 + 6.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.20 + 1.39i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 0.285T + 89T^{2} \)
97 \( 1 + (8.73 - 2.34i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.04580571529234092046130939241, −12.04285928300442235432978478569, −10.21185437569308861284349878683, −9.353278444438021213701318297880, −8.283337645854978520035932713220, −7.54956273627400049581069756337, −6.70837480756673521368474613403, −5.35829593492900687252415874080, −4.34772525559798335854197580137, −2.85884297667175772898085992997, 1.64883436630750658534237352550, 2.93811372769568864298938274970, 3.84290175615894868704242199115, 5.08497462949695681930811128757, 6.97041211315123173893848614866, 8.199002224184645261489338779808, 9.446544866617336754016484562700, 9.965010703769972517949931394471, 10.93015763900430112868584160135, 12.21995715473132869947173517408

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