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Properties

Label 2-15e2-9.7-c3-0-48
Degree $2$
Conductor $225$
Sign $-0.993 - 0.112i$
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  + (2.18 − 3.78i)2-s + (−3.55 + 3.78i)3-s + (−5.55 − 9.62i)4-s + (6.55 + 21.7i)6-s + (6.05 − 10.4i)7-s − 13.6·8-s + (−1.67 − 26.9i)9-s + (−5.01 + 8.67i)11-s + (56.2 + 13.2i)12-s + (−24.2 − 42.0i)13-s + (−26.4 − 45.8i)14-s + (14.6 − 25.4i)16-s − 75.3·17-s + (−105. − 52.5i)18-s − 116.·19-s + ⋯
L(s)  = 1  + (0.772 − 1.33i)2-s + (−0.684 + 0.728i)3-s + (−0.694 − 1.20i)4-s + (0.446 + 1.48i)6-s + (0.327 − 0.566i)7-s − 0.602·8-s + (−0.0620 − 0.998i)9-s + (−0.137 + 0.237i)11-s + (1.35 + 0.317i)12-s + (−0.518 − 0.897i)13-s + (−0.505 − 0.875i)14-s + (0.229 − 0.397i)16-s − 1.07·17-s + (−1.38 − 0.688i)18-s − 1.40·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.993 - 0.112i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.993 - 0.112i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0721760 + 1.28239i\)
\(L(\frac12)\) \(\approx\) \(0.0721760 + 1.28239i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.55 - 3.78i)T \)
5 \( 1 \)
good2 \( 1 + (-2.18 + 3.78i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-6.05 + 10.4i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (5.01 - 8.67i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (24.2 + 42.0i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 75.3T + 4.91e3T^{2} \)
19 \( 1 + 116.T + 6.85e3T^{2} \)
23 \( 1 + (19.0 + 32.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-11.3 + 19.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (15.0 + 26.0i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 130.T + 5.06e4T^{2} \)
41 \( 1 + (173. + 300. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-13.3 + 23.1i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-230. + 399. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 438.T + 1.48e5T^{2} \)
59 \( 1 + (4.18 + 7.24i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-41.0 + 71.0i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-341. - 591. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.09e3T + 3.57e5T^{2} \)
73 \( 1 + 470.T + 3.89e5T^{2} \)
79 \( 1 + (243. - 420. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-49.5 + 85.8i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 8.80T + 7.04e5T^{2} \)
97 \( 1 + (-330. + 572. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.21114661564349237302599206059, −10.54373658996200876412811899075, −10.04432952183238208003178331409, −8.658316301553747349617489620930, −7.01014074070585107395954132915, −5.53672978439798156494304621020, −4.56657946510444561183916537035, −3.80950521793689328023317583248, −2.28462096794983056226665382252, −0.42827649761870618714149665032, 2.07114951500847714136711809746, 4.35447940334998705043850099933, 5.22920726812040294575500129780, 6.29302494718641980761813360486, 6.88042029394192880334161548911, 7.980607348970154009652859227216, 8.868962910386438345700527680954, 10.62481019900230282873714396546, 11.61954848493616239515166215317, 12.55870330373898200990361731362

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