Properties

Label 2-15e2-225.106-c1-0-5
Degree $2$
Conductor $225$
Sign $0.881 - 0.472i$
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  + (−2.47 − 0.526i)2-s + (−1.29 + 1.15i)3-s + (4.03 + 1.79i)4-s + (1.67 − 1.48i)5-s + (3.81 − 2.16i)6-s + (−1.03 + 1.79i)7-s + (−4.94 − 3.59i)8-s + (0.350 − 2.97i)9-s + (−4.92 + 2.80i)10-s + (−2.39 − 0.509i)11-s + (−7.28 + 2.31i)12-s + (1.74 − 0.371i)13-s + (3.50 − 3.89i)14-s + (−0.452 + 3.84i)15-s + (4.44 + 4.94i)16-s + (3.67 + 2.66i)17-s + ⋯
L(s)  = 1  + (−1.75 − 0.372i)2-s + (−0.747 + 0.664i)3-s + (2.01 + 0.897i)4-s + (0.747 − 0.664i)5-s + (1.55 − 0.885i)6-s + (−0.390 + 0.676i)7-s + (−1.74 − 1.26i)8-s + (0.116 − 0.993i)9-s + (−1.55 + 0.885i)10-s + (−0.722 − 0.153i)11-s + (−2.10 + 0.668i)12-s + (0.484 − 0.102i)13-s + (0.936 − 1.04i)14-s + (−0.116 + 0.993i)15-s + (1.11 + 1.23i)16-s + (0.891 + 0.647i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.881 - 0.472i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.881 - 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.456025 + 0.114427i\)
\(L(\frac12)\) \(\approx\) \(0.456025 + 0.114427i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.29 - 1.15i)T \)
5 \( 1 + (-1.67 + 1.48i)T \)
good2 \( 1 + (2.47 + 0.526i)T + (1.82 + 0.813i)T^{2} \)
7 \( 1 + (1.03 - 1.79i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.39 + 0.509i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-1.74 + 0.371i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (-3.67 - 2.66i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-5.85 - 4.25i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.58 + 1.75i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (-0.855 - 8.13i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (0.709 - 6.74i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-1.53 + 4.73i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-4.68 + 0.995i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-4.14 + 7.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.05 - 10.0i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (2.52 - 1.83i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (10.2 - 2.17i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-2.57 - 0.548i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (-0.170 + 1.62i)T + (-65.5 - 13.9i)T^{2} \)
71 \( 1 + (-8.24 + 5.98i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.608 + 1.87i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.55 + 14.8i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-3.70 + 1.65i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-0.105 - 0.326i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.15 - 10.9i)T + (-94.8 + 20.1i)T^{2} \)
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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

−12.25111556557661049973649842053, −10.79499701896827361778627809002, −10.37264379734685469245952919670, −9.362989314057613446765116276665, −8.885997463423283558030770654234, −7.67510081974011972612317841798, −6.17140885690663740883443589922, −5.34249557875273612518377990678, −3.13801780513246908978135179245, −1.24924017804354877153219695777, 0.905369034657321267552485791033, 2.57290968824194517097890269377, 5.48882783992019169058045351198, 6.46974501799522182227595865812, 7.30442377893333359271848317523, 7.894604997789627436702431697799, 9.572998604859615837044511625731, 9.979180100663370330183351105602, 11.02571109590368636374857653269, 11.57480147460216937370261901147

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