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Properties

Label 2-15e2-45.4-c1-0-8
Degree $2$
Conductor $225$
Sign $0.999 - 0.0426i$
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  + (0.495 − 0.285i)2-s + (1.70 − 0.285i)3-s + (−0.836 + 1.44i)4-s + (0.764 − 0.630i)6-s + (1.23 − 0.714i)7-s + 2.10i·8-s + (2.83 − 0.977i)9-s + (−1.33 − 2.31i)11-s + (−1.01 + 2.71i)12-s + (4.04 + 2.33i)13-s + (0.408 − 0.707i)14-s + (−1.07 − 1.85i)16-s + 2.67i·17-s + (1.12 − 1.29i)18-s − 4.67·19-s + ⋯
L(s)  = 1  + (0.350 − 0.202i)2-s + (0.986 − 0.165i)3-s + (−0.418 + 0.724i)4-s + (0.312 − 0.257i)6-s + (0.467 − 0.269i)7-s + 0.742i·8-s + (0.945 − 0.325i)9-s + (−0.402 − 0.697i)11-s + (−0.292 + 0.783i)12-s + (1.12 + 0.648i)13-s + (0.109 − 0.189i)14-s + (−0.267 − 0.464i)16-s + 0.648i·17-s + (0.265 − 0.305i)18-s − 1.07·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.999 - 0.0426i$
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: \((2,\ 225,\ (\ :1/2),\ 0.999 - 0.0426i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82672 + 0.0389564i\)
\(L(\frac12)\) \(\approx\) \(1.82672 + 0.0389564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.285i)T \)
5 \( 1 \)
good2 \( 1 + (-0.495 + 0.285i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-1.23 + 0.714i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.33 + 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.04 - 2.33i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.67iT - 17T^{2} \)
19 \( 1 + 4.67T + 19T^{2} \)
23 \( 1 + (5.12 + 2.95i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.74 + 8.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.48 - 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.81iT - 37T^{2} \)
41 \( 1 + (-0.735 + 1.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.408 - 0.235i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.02 + 3.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.14iT - 53T^{2} \)
59 \( 1 + (0.571 - 0.990i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.26 - 2.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.70 - 3.29i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 - 1.71iT - 73T^{2} \)
79 \( 1 + (0.143 + 0.249i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.71 + 2.14i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-6.78 + 3.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.52812579412671848286397462564, −11.38861859389619236624472679248, −10.39965612360711794830835853080, −8.906665649654667935131324649327, −8.420180214099384432096097293896, −7.56509582437210208667474433412, −6.09994461879477874880336056482, −4.32883913078736191418348335676, −3.64853079067432313172858473617, −2.13285895480032997415493496905, 1.87336085586554893542473721574, 3.67037633533474885704514514490, 4.78265042715602144012727159352, 5.89466141351110035527182863038, 7.31093148873255507496347165610, 8.384456398448423474109165749969, 9.273277887571899767906279140623, 10.17978775280050790014940566109, 11.06793572692650064203449110140, 12.67763222971671493366902988467

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