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Properties

Label 2-15e2-45.34-c3-0-21
Degree $2$
Conductor $225$
Sign $0.973 - 0.228i$
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  + (−0.151 − 0.0874i)2-s + (−0.151 + 5.19i)3-s + (−3.98 − 6.90i)4-s + (0.477 − 0.773i)6-s + (7.32 + 4.23i)7-s + 2.79i·8-s + (−26.9 − 1.57i)9-s + (15.7 − 27.2i)11-s + (36.4 − 19.6i)12-s + (23.2 − 13.4i)13-s + (−0.740 − 1.28i)14-s + (−31.6 + 54.7i)16-s + 44.3i·17-s + (3.94 + 2.59i)18-s + 90.2·19-s + ⋯
L(s)  = 1  + (−0.0535 − 0.0309i)2-s + (−0.0291 + 0.999i)3-s + (−0.498 − 0.862i)4-s + (0.0324 − 0.0526i)6-s + (0.395 + 0.228i)7-s + 0.123i·8-s + (−0.998 − 0.0583i)9-s + (0.431 − 0.747i)11-s + (0.876 − 0.472i)12-s + (0.496 − 0.286i)13-s + (−0.0141 − 0.0244i)14-s + (−0.494 + 0.856i)16-s + 0.632i·17-s + (0.0516 + 0.0340i)18-s + 1.08·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.973 - 0.228i$
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: \((2,\ 225,\ (\ :3/2),\ 0.973 - 0.228i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.58433 + 0.183631i\)
\(L(\frac12)\) \(\approx\) \(1.58433 + 0.183631i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.151 - 5.19i)T \)
5 \( 1 \)
good2 \( 1 + (0.151 + 0.0874i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (-7.32 - 4.23i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-15.7 + 27.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-23.2 + 13.4i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 44.3iT - 4.91e3T^{2} \)
19 \( 1 - 90.2T + 6.85e3T^{2} \)
23 \( 1 + (-168. + 97.1i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-1.87 + 3.24i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-125. - 217. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 62.2iT - 5.06e4T^{2} \)
41 \( 1 + (-102. - 176. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-456. - 263. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-134. - 77.8i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 141. iT - 1.48e5T^{2} \)
59 \( 1 + (246. + 427. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-379. + 657. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-470. + 271. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 928.T + 3.57e5T^{2} \)
73 \( 1 - 608. iT - 3.89e5T^{2} \)
79 \( 1 + (-307. + 532. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (931. + 537. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + (288. + 166. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.37771666175995134542728575086, −10.86432255121011897740347599123, −9.899503019003838300592318313958, −8.976472964167765151004054274727, −8.319362286700637397790265477033, −6.38206723362575935095169954598, −5.42192865655820071707983198447, −4.53469917261325344791050882161, −3.15971426531087709991911783470, −0.991676195477600213502830801949, 1.02843462551580343210086742203, 2.75444455338161256012979145838, 4.19617663442034066250528517354, 5.55370507782768110016127214325, 7.17465565256331279209198153908, 7.47731489119058934872186818249, 8.741703398399831161056889485661, 9.484104381095577592203910686915, 11.19493623057326917028730244431, 11.86694019221883000824242776105

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