Properties

Label 2-15e2-25.6-c3-0-11
Degree $2$
Conductor $225$
Sign $-0.620 - 0.783i$
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  + (1.20 + 3.70i)2-s + (−5.80 + 4.22i)4-s + (−3.34 − 10.6i)5-s + 6.27·7-s + (2.58 + 1.87i)8-s + (35.5 − 25.2i)10-s + (13.6 + 41.9i)11-s + (−13.2 + 40.7i)13-s + (7.56 + 23.2i)14-s + (−21.5 + 66.4i)16-s + (81.6 + 59.3i)17-s + (−65.1 − 47.3i)19-s + (64.4 + 47.8i)20-s + (−139. + 101. i)22-s + (48.5 + 149. i)23-s + ⋯
L(s)  = 1  + (0.425 + 1.31i)2-s + (−0.726 + 0.527i)4-s + (−0.298 − 0.954i)5-s + 0.339·7-s + (0.114 + 0.0829i)8-s + (1.12 − 0.797i)10-s + (0.373 + 1.15i)11-s + (−0.282 + 0.870i)13-s + (0.144 + 0.444i)14-s + (−0.337 + 1.03i)16-s + (1.16 + 0.846i)17-s + (−0.786 − 0.571i)19-s + (0.720 + 0.535i)20-s + (−1.34 + 0.979i)22-s + (0.440 + 1.35i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.620 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.620 - 0.783i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.620 - 0.783i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.948544 + 1.96110i\)
\(L(\frac12)\) \(\approx\) \(0.948544 + 1.96110i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (3.34 + 10.6i)T \)
good2 \( 1 + (-1.20 - 3.70i)T + (-6.47 + 4.70i)T^{2} \)
7 \( 1 - 6.27T + 343T^{2} \)
11 \( 1 + (-13.6 - 41.9i)T + (-1.07e3 + 782. i)T^{2} \)
13 \( 1 + (13.2 - 40.7i)T + (-1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (-81.6 - 59.3i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (65.1 + 47.3i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + (-48.5 - 149. i)T + (-9.84e3 + 7.15e3i)T^{2} \)
29 \( 1 + (-199. + 145. i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (-69.6 - 50.6i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (-61.0 + 188. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (17.2 - 53.1i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 - 211.T + 7.95e4T^{2} \)
47 \( 1 + (85.9 - 62.4i)T + (3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (-153. + 111. i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-140. + 432. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (-171. - 528. i)T + (-1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 + (773. + 561. i)T + (9.29e4 + 2.86e5i)T^{2} \)
71 \( 1 + (454. - 330. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (54.7 + 168. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (769. - 558. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (11.3 + 8.23i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 + (-143. - 440. i)T + (-5.70e5 + 4.14e5i)T^{2} \)
97 \( 1 + (-1.05e3 + 764. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−12.34811741582627722701168468558, −11.43893636956918330986200555939, −9.950854002159003589833419637244, −8.875451200023508241307201195704, −7.924941376113002258665742205485, −7.12552213418623125507343279485, −5.97294084964647814491549035863, −4.83729311908622822391114177954, −4.17931864941140793077656949427, −1.64008005518493358562630872603, 0.858080805123662528769977544782, 2.69665551328559042881468661703, 3.37958191661285017332028540018, 4.71919950510029094974219301104, 6.18399134047686036663523569054, 7.47101887102277400708806514869, 8.574682494867938828800998221923, 10.16982633154202126724079091209, 10.54511597501999833087773267888, 11.52624487743885926522348550884

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