Properties

Label 2-15e2-45.38-c1-0-14
Degree $2$
Conductor $225$
Sign $0.565 + 0.824i$
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  + (1.60 − 0.430i)2-s + (1.08 − 1.35i)3-s + (0.661 − 0.382i)4-s + (1.15 − 2.63i)6-s + (−0.465 − 1.73i)7-s + (−1.45 + 1.45i)8-s + (−0.661 − 2.92i)9-s + (3.12 + 1.80i)11-s + (0.198 − 1.30i)12-s + (−0.342 + 1.27i)13-s + (−1.49 − 2.59i)14-s + (−2.47 + 4.28i)16-s + (0.277 + 0.277i)17-s + (−2.32 − 4.41i)18-s + 6.25i·19-s + ⋯
L(s)  = 1  + (1.13 − 0.304i)2-s + (0.624 − 0.781i)3-s + (0.330 − 0.191i)4-s + (0.471 − 1.07i)6-s + (−0.175 − 0.656i)7-s + (−0.513 + 0.513i)8-s + (−0.220 − 0.975i)9-s + (0.942 + 0.544i)11-s + (0.0573 − 0.377i)12-s + (−0.0950 + 0.354i)13-s + (−0.399 − 0.692i)14-s + (−0.618 + 1.07i)16-s + (0.0671 + 0.0671i)17-s + (−0.547 − 1.04i)18-s + 1.43i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.565 + 0.824i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.565 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03061 - 1.06914i\)
\(L(\frac12)\) \(\approx\) \(2.03061 - 1.06914i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.08 + 1.35i)T \)
5 \( 1 \)
good2 \( 1 + (-1.60 + 0.430i)T + (1.73 - i)T^{2} \)
7 \( 1 + (0.465 + 1.73i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.12 - 1.80i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.342 - 1.27i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.277 - 0.277i)T + 17iT^{2} \)
19 \( 1 - 6.25iT - 19T^{2} \)
23 \( 1 + (2.16 + 0.579i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.56 + 2.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.42 + 4.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.55 - 5.55i)T - 37iT^{2} \)
41 \( 1 + (-1.29 + 0.744i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.10 - 1.10i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.82 + 1.02i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.48 + 7.48i)T - 53iT^{2} \)
59 \( 1 + (0.279 + 0.483i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.96 - 5.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.8 + 2.90i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.01iT - 71T^{2} \)
73 \( 1 + (-1.29 - 1.29i)T + 73iT^{2} \)
79 \( 1 + (6.96 + 4.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.150 - 0.560i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 + (-1.37 - 5.14i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.10941729814631836935164303597, −11.84101187146025711660980316555, −10.21783719566277684684332981895, −9.095627859699502585609575689380, −8.019743908235567682314112967779, −6.87202145797141210086146084403, −5.91240187913987218925282310765, −4.26402657859701556839653472452, −3.48998377490408359748473657877, −1.90181723413882924293658924398, 2.82969006167080384891745720267, 3.84956480333767904856857731471, 4.96857292238856122986138619989, 5.88240479221608503386603413874, 7.14035037488592814659373013408, 8.803679109713754653709094897272, 9.200182343168605268804869180227, 10.48041154819459374579009196084, 11.64968826736465444811850735698, 12.59153205551665025320276603578

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