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Properties

Label 2-15e2-9.4-c3-0-30
Degree $2$
Conductor $225$
Sign $0.401 - 0.915i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.23 + 2.13i)2-s + (4.97 + 1.50i)3-s + (0.964 − 1.67i)4-s + (2.91 + 12.4i)6-s + (9.62 + 16.6i)7-s + 24.4·8-s + (22.4 + 14.9i)9-s + (−19.9 − 34.5i)11-s + (7.31 − 6.85i)12-s + (−0.504 + 0.873i)13-s + (−23.7 + 41.0i)14-s + (22.4 + 38.8i)16-s + 52.6·17-s + (−4.27 + 66.3i)18-s − 49.5·19-s + ⋯
L(s)  = 1  + (0.435 + 0.754i)2-s + (0.957 + 0.289i)3-s + (0.120 − 0.208i)4-s + (0.198 + 0.848i)6-s + (0.519 + 0.899i)7-s + 1.08·8-s + (0.832 + 0.554i)9-s + (−0.546 − 0.946i)11-s + (0.175 − 0.164i)12-s + (−0.0107 + 0.0186i)13-s + (−0.452 + 0.783i)14-s + (0.350 + 0.606i)16-s + 0.750·17-s + (−0.0559 + 0.869i)18-s − 0.598·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.401 - 0.915i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.401 - 0.915i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.99774 + 1.95846i\)
\(L(\frac12)\) \(\approx\) \(2.99774 + 1.95846i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.97 - 1.50i)T \)
5 \( 1 \)
good2 \( 1 + (-1.23 - 2.13i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-9.62 - 16.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (19.9 + 34.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (0.504 - 0.873i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 52.6T + 4.91e3T^{2} \)
19 \( 1 + 49.5T + 6.85e3T^{2} \)
23 \( 1 + (13.7 - 23.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-127. - 220. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-84.3 + 146. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 419.T + 5.06e4T^{2} \)
41 \( 1 + (199. - 345. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (179. + 310. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (70.6 + 122. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 290.T + 1.48e5T^{2} \)
59 \( 1 + (14.3 - 24.8i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (366. + 634. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-88.2 + 152. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 802.T + 3.57e5T^{2} \)
73 \( 1 - 512.T + 3.89e5T^{2} \)
79 \( 1 + (306. + 530. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (40.4 + 69.9i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 24.0T + 7.04e5T^{2} \)
97 \( 1 + (683. + 1.18e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.10427765996643432514554130479, −10.83437538225091326246227886040, −10.03942913861727284553153527041, −8.657400472583569498308592370286, −8.119335677413779602842992628709, −6.91483830250644676501440004832, −5.63225374423313061604799423603, −4.82096576681043425200845727770, −3.24572184027379445513649677102, −1.79397861492162687117973202231, 1.45743824212775788053814227198, 2.62044535670083590863201648316, 3.85774145704757772861286673796, 4.75171344416654058105249071081, 6.89286825652210393263098177737, 7.65921973988646111334331993522, 8.460095714791514838221981390170, 10.07350107527731383021700812833, 10.49897708512518260670487833779, 11.93006150065995896064409763171

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