Properties

Label 2-15e2-9.7-c3-0-17
Degree $2$
Conductor $225$
Sign $-0.813 - 0.581i$
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  + (−2.13 + 3.69i)2-s + (−4.39 + 2.76i)3-s + (−5.08 − 8.80i)4-s + (−0.854 − 22.1i)6-s + (15.3 − 26.6i)7-s + 9.21·8-s + (11.6 − 24.3i)9-s + (−20.3 + 35.2i)11-s + (46.7 + 24.6i)12-s + (31.6 + 54.7i)13-s + (65.5 + 113. i)14-s + (21.0 − 36.3i)16-s + 6.58·17-s + (65.0 + 94.9i)18-s + 75.3·19-s + ⋯
L(s)  = 1  + (−0.753 + 1.30i)2-s + (−0.846 + 0.533i)3-s + (−0.635 − 1.10i)4-s + (−0.0581 − 1.50i)6-s + (0.830 − 1.43i)7-s + 0.407·8-s + (0.431 − 0.902i)9-s + (−0.557 + 0.966i)11-s + (1.12 + 0.592i)12-s + (0.674 + 1.16i)13-s + (1.25 + 2.16i)14-s + (0.328 − 0.568i)16-s + 0.0940·17-s + (0.851 + 1.24i)18-s + 0.910·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.813 - 0.581i$
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: \((2,\ 225,\ (\ :3/2),\ -0.813 - 0.581i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.247794 + 0.772312i\)
\(L(\frac12)\) \(\approx\) \(0.247794 + 0.772312i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.39 - 2.76i)T \)
5 \( 1 \)
good2 \( 1 + (2.13 - 3.69i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-15.3 + 26.6i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (20.3 - 35.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-31.6 - 54.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 6.58T + 4.91e3T^{2} \)
19 \( 1 - 75.3T + 6.85e3T^{2} \)
23 \( 1 + (31.1 + 54.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (24.8 - 42.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (51.5 + 89.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 282.T + 5.06e4T^{2} \)
41 \( 1 + (-78.7 - 136. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (168. - 292. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-22.2 + 38.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 26.2T + 1.48e5T^{2} \)
59 \( 1 + (-212. - 368. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (425. - 736. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-48.1 - 83.4i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 952.T + 3.57e5T^{2} \)
73 \( 1 - 50.8T + 3.89e5T^{2} \)
79 \( 1 + (-98.6 + 170. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (98.8 - 171. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + (715. - 1.23e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.89507741131969824252853537995, −10.99417869145272286424349232605, −10.05505117213232459000370407193, −9.287021728531581622230413226235, −7.909004684028513948184403449626, −7.19537468864474324748851260313, −6.30836649586687089196022566433, −5.02002949622101884899089453123, −4.13369256444673146217363568054, −1.04937979076290453807642734220, 0.62795536976154033942543766677, 1.92708502888368088453311177365, 3.16157489309598086665766024539, 5.31847326183973496045579051624, 5.92645009508196190190065747752, 7.900073746093623628758692345983, 8.465709679518752526139590984837, 9.667023064320936434438485343683, 10.86792188132860499311604344658, 11.24717777227432531191989593204

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