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Properties

Label 2-15e2-45.29-c2-0-2
Degree $2$
Conductor $225$
Sign $-0.687 + 0.726i$
Analytic cond. $6.13080$
Root an. cond. $2.47604$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 1.5i)2-s + (−2.59 − 1.5i)3-s + (0.5 + 0.866i)4-s + (4.5 − 2.59i)6-s + (1.73 + i)7-s − 8.66·8-s + (4.5 + 7.79i)9-s + (−1.5 − 0.866i)11-s − 2.99i·12-s + (−3.46 + 2i)13-s + (−3 + 1.73i)14-s + (5.5 − 9.52i)16-s − 15.5·17-s − 15.5·18-s − 11·19-s + ⋯
L(s)  = 1  + (−0.433 + 0.750i)2-s + (−0.866 − 0.5i)3-s + (0.125 + 0.216i)4-s + (0.750 − 0.433i)6-s + (0.247 + 0.142i)7-s − 1.08·8-s + (0.5 + 0.866i)9-s + (−0.136 − 0.0787i)11-s − 0.249i·12-s + (−0.266 + 0.153i)13-s + (−0.214 + 0.123i)14-s + (0.343 − 0.595i)16-s − 0.916·17-s − 0.866·18-s − 0.578·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.687 + 0.726i$
Analytic conductor: \(6.13080\)
Root analytic conductor: \(2.47604\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1),\ -0.687 + 0.726i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0226763 - 0.0526982i\)
\(L(\frac12)\) \(\approx\) \(0.0226763 - 0.0526982i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.59 + 1.5i)T \)
5 \( 1 \)
good2 \( 1 + (0.866 - 1.5i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (-1.73 - i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.46 - 2i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 15.5T + 289T^{2} \)
19 \( 1 + 11T + 361T^{2} \)
23 \( 1 + (13.8 + 24i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (39 + 22.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (16 + 27.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 34iT - 1.36e3T^{2} \)
41 \( 1 + (10.5 - 6.06i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-52.8 - 30.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (24.2 - 42i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + (43.5 - 25.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (28 - 48.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-26.8 + 15.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 31.1iT - 5.04e3T^{2} \)
73 \( 1 - 65iT - 5.32e3T^{2} \)
79 \( 1 + (-19 + 32.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-24.2 + 42i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 124. iT - 7.92e3T^{2} \)
97 \( 1 + (99.5 + 57.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.55755183575513068912311968038, −11.63462319605933845978381339870, −10.90895882114158059459822781322, −9.566392860328901170113385413407, −8.367412209557000427900939142448, −7.57271654216396163357351416880, −6.57320728183164972682665553067, −5.83038551952258134801018397554, −4.39270447797196656570702721761, −2.28962656638144170372561036126, 0.03739152028739297915017751937, 1.82650018701440605104984158172, 3.65428756780633052740021703406, 5.09372193097662991698157150632, 6.08688779296623637405323482038, 7.26124969701482941042838466002, 8.922783260412090620466039097658, 9.669674731859542180085565914099, 10.75048771931182732696089667463, 11.05210454100594760533922280347

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