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Properties

Label 2-15e2-9.7-c1-0-13
Degree $2$
Conductor $225$
Sign $-0.642 + 0.766i$
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  + (0.965 − 1.67i)2-s + (0.448 − 1.67i)3-s + (−0.866 − 1.50i)4-s + (−2.36 − 2.36i)6-s + (0.448 − 0.776i)7-s + 0.517·8-s + (−2.59 − 1.50i)9-s + (−2.36 + 4.09i)11-s + (−2.89 + 0.776i)12-s + (1.22 + 2.12i)13-s + (−0.866 − 1.50i)14-s + (2.23 − 3.86i)16-s − 0.378·17-s + (−5.01 + 2.89i)18-s + 2.73·19-s + ⋯
L(s)  = 1  + (0.683 − 1.18i)2-s + (0.258 − 0.965i)3-s + (−0.433 − 0.750i)4-s + (−0.965 − 0.965i)6-s + (0.169 − 0.293i)7-s + 0.183·8-s + (−0.866 − 0.5i)9-s + (−0.713 + 1.23i)11-s + (−0.836 + 0.224i)12-s + (0.339 + 0.588i)13-s + (−0.231 − 0.400i)14-s + (0.558 − 0.966i)16-s − 0.0919·17-s + (−1.18 + 0.683i)18-s + 0.626·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778579 - 1.66966i\)
\(L(\frac12)\) \(\approx\) \(0.778579 - 1.66966i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.448 + 1.67i)T \)
5 \( 1 \)
good2 \( 1 + (-0.965 + 1.67i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.448 + 0.776i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.36 - 4.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.378T + 17T^{2} \)
19 \( 1 - 2.73T + 19T^{2} \)
23 \( 1 + (0.965 + 1.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.23 - 5.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.36 - 2.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + (2.13 + 3.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.57 + 7.91i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.19 + 3.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.86T + 53T^{2} \)
59 \( 1 + (1.26 + 2.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.33 - 9.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.56 + 4.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.80T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 + (0.267 - 0.464i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.20 - 9.02i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.39T + 89T^{2} \)
97 \( 1 + (5.13 - 8.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.18464990268474444883838611646, −11.18751380192326378627967852105, −10.30617146173183993303840354110, −9.107085653308739837138071339387, −7.72169736519183307265148829915, −6.97983370216034175881455624521, −5.35697285724888169395945063147, −4.09678938218898410482516246316, −2.71141039109583131863725020263, −1.58788031483045122153443687870, 3.06501694214300824385375660298, 4.34512017051328941468936154739, 5.50942299647231440766772225672, 6.01771676475099251094094272467, 7.75706460371404884367238788537, 8.311918729733644591706597827928, 9.568132573878319405459373831269, 10.70378556208389817292052249119, 11.47853628800358665117095130253, 13.12267142224952845880851700397

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