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Properties

Label 2-15e2-45.4-c3-0-32
Degree $2$
Conductor $225$
Sign $0.405 + 0.914i$
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  + (−3.24 + 1.87i)2-s + (−3.24 − 4.05i)3-s + (3.02 − 5.24i)4-s + (18.1 + 7.08i)6-s + (27.1 − 15.6i)7-s − 7.30i·8-s + (−5.92 + 26.3i)9-s + (10.4 + 18.0i)11-s + (−31.0 + 4.73i)12-s + (−51.9 − 29.9i)13-s + (−58.7 + 101. i)14-s + (37.8 + 65.6i)16-s + 74.0i·17-s + (−30.1 − 96.6i)18-s + 63.8·19-s + ⋯
L(s)  = 1  + (−1.14 + 0.662i)2-s + (−0.624 − 0.780i)3-s + (0.378 − 0.655i)4-s + (1.23 + 0.482i)6-s + (1.46 − 0.846i)7-s − 0.322i·8-s + (−0.219 + 0.975i)9-s + (0.285 + 0.494i)11-s + (−0.747 + 0.113i)12-s + (−1.10 − 0.639i)13-s + (−1.12 + 1.94i)14-s + (0.592 + 1.02i)16-s + 1.05i·17-s + (−0.394 − 1.26i)18-s + 0.770·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.405 + 0.914i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.405 + 0.914i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.593022 - 0.385891i\)
\(L(\frac12)\) \(\approx\) \(0.593022 - 0.385891i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.24 + 4.05i)T \)
5 \( 1 \)
good2 \( 1 + (3.24 - 1.87i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (-27.1 + 15.6i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-10.4 - 18.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (51.9 + 29.9i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 74.0iT - 4.91e3T^{2} \)
19 \( 1 - 63.8T + 6.85e3T^{2} \)
23 \( 1 + (-28.4 - 16.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (80.0 + 138. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-127. + 220. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 215. iT - 5.06e4T^{2} \)
41 \( 1 + (70.8 - 122. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-119. + 68.9i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (29.0 - 16.7i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 41.9iT - 1.48e5T^{2} \)
59 \( 1 + (-307. + 532. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-67.1 - 116. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (742. + 428. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 588.T + 3.57e5T^{2} \)
73 \( 1 + 618. iT - 3.89e5T^{2} \)
79 \( 1 + (172. + 299. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (946. - 546. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 414.T + 7.04e5T^{2} \)
97 \( 1 + (-174. + 100. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.44637910194286520151470285842, −10.55162727790593677387055468480, −9.658085246418327916446413491634, −8.131006410246310506542025868503, −7.72864938084019278294532490000, −6.98759522951534803491759936105, −5.67173272888122824496574119372, −4.38545776763668880380813152235, −1.77789298359951485185831749657, −0.55100565006499660486143982023, 1.21423754510935957614729981842, 2.76601589535534873282365207288, 4.79276307730672957681103091369, 5.40661251267145946922945889921, 7.19527834211608981168615297889, 8.531823571843959064804759070826, 9.142302186598407559889966930041, 10.02368954462386702184698401067, 11.02242292215592713082494782264, 11.73236012890419271272299246723

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