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Properties

Label 2-15e2-45.34-c3-0-41
Degree $2$
Conductor $225$
Sign $0.467 + 0.883i$
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  + (2.63 + 1.52i)2-s + (−3.29 + 4.01i)3-s + (0.626 + 1.08i)4-s + (−14.7 + 5.57i)6-s + (−11.8 − 6.85i)7-s − 20.5i·8-s + (−5.29 − 26.4i)9-s + (−15.9 + 27.5i)11-s + (−6.42 − 1.05i)12-s + (50.4 − 29.1i)13-s + (−20.8 − 36.1i)14-s + (36.2 − 62.7i)16-s − 109. i·17-s + (26.3 − 77.8i)18-s − 129.·19-s + ⋯
L(s)  = 1  + (0.931 + 0.537i)2-s + (−0.633 + 0.773i)3-s + (0.0782 + 0.135i)4-s + (−1.00 + 0.379i)6-s + (−0.641 − 0.370i)7-s − 0.907i·8-s + (−0.196 − 0.980i)9-s + (−0.435 + 0.755i)11-s + (−0.154 − 0.0254i)12-s + (1.07 − 0.621i)13-s + (−0.398 − 0.689i)14-s + (0.566 − 0.980i)16-s − 1.55i·17-s + (0.344 − 1.01i)18-s − 1.56·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.467 + 0.883i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.467 + 0.883i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.07533 - 0.647607i\)
\(L(\frac12)\) \(\approx\) \(1.07533 - 0.647607i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.29 - 4.01i)T \)
5 \( 1 \)
good2 \( 1 + (-2.63 - 1.52i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (11.8 + 6.85i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (15.9 - 27.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-50.4 + 29.1i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 109. iT - 4.91e3T^{2} \)
19 \( 1 + 129.T + 6.85e3T^{2} \)
23 \( 1 + (-68.9 + 39.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-4.51 + 7.82i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-16.6 - 28.9i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 22.1iT - 5.06e4T^{2} \)
41 \( 1 + (60.8 + 105. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (8.78 + 5.07i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (382. + 220. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 593. iT - 1.48e5T^{2} \)
59 \( 1 + (-221. - 383. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (72.2 - 125. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (747. - 431. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 818.T + 3.57e5T^{2} \)
73 \( 1 - 495. iT - 3.89e5T^{2} \)
79 \( 1 + (-585. + 1.01e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-367. - 212. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.03e3T + 7.04e5T^{2} \)
97 \( 1 + (-1.38e3 - 799. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.73371310334102837171353624750, −10.51810162756067420942863137781, −9.959711413865632045259112007993, −8.807137309249332987725945361424, −7.06233768549379922792612059456, −6.30840295159690824506579001292, −5.23239693189462034607771138778, −4.39648356691820881684715434670, −3.27462001484702289422355335612, −0.40934228026168924337893116820, 1.76926175662941319782700683417, 3.18805112400585211050543065343, 4.46289616132967422334309236466, 5.89690215925446374947813269995, 6.34768085807500236323258762588, 8.051725756249062417512850541970, 8.799232762604976198431791313302, 10.67930259830503981665126575818, 11.13729116971157972927788910102, 12.19028029473862260602012441998

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