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Properties

Label 2-15e2-45.34-c1-0-8
Degree $2$
Conductor $225$
Sign $-0.551 + 0.833i$
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  + (−2.17 − 1.25i)2-s + (1.19 − 1.25i)3-s + (2.16 + 3.74i)4-s + (−4.17 + 1.23i)6-s + (0.445 + 0.257i)7-s − 5.83i·8-s + (−0.160 − 2.99i)9-s + (1.66 − 2.87i)11-s + (7.27 + 1.74i)12-s + (1.14 − 0.660i)13-s + (−0.646 − 1.11i)14-s + (−3.01 + 5.22i)16-s − 3.32i·17-s + (−3.41 + 6.72i)18-s + 1.32·19-s + ⋯
L(s)  = 1  + (−1.53 − 0.888i)2-s + (0.687 − 0.725i)3-s + (1.08 + 1.87i)4-s + (−1.70 + 0.505i)6-s + (0.168 + 0.0971i)7-s − 2.06i·8-s + (−0.0534 − 0.998i)9-s + (0.500 − 0.867i)11-s + (2.10 + 0.503i)12-s + (0.317 − 0.183i)13-s + (−0.172 − 0.299i)14-s + (−0.753 + 1.30i)16-s − 0.805i·17-s + (−0.805 + 1.58i)18-s + 0.303·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.551 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.551 + 0.833i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.551 + 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.351815 - 0.654752i\)
\(L(\frac12)\) \(\approx\) \(0.351815 - 0.654752i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.19 + 1.25i)T \)
5 \( 1 \)
good2 \( 1 + (2.17 + 1.25i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-0.445 - 0.257i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.66 + 2.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.14 + 0.660i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.32iT - 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + (3.57 - 2.06i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.693 - 1.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.36 + 7.56i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.292iT - 37T^{2} \)
41 \( 1 + (-5.67 - 9.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.96 - 5.17i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.21 - 2.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.02iT - 53T^{2} \)
59 \( 1 + (2.51 + 4.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.67 - 6.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.18 + 4.72i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.99T + 71T^{2} \)
73 \( 1 + 6.05iT - 73T^{2} \)
79 \( 1 + (4.02 - 6.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.33 - 0.771i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-10.6 - 6.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.61872241512772223894340806119, −11.04737368879577163051152661460, −9.609336069632407756861126154678, −9.136367527909812081463677652483, −8.088949764097681019029144080561, −7.50503882073280276853271340902, −6.14563106304669760985894017212, −3.61518848473022671311554222943, −2.45914750752995271755829095510, −1.01904915821286136086626410457, 1.90213756689177971949256658658, 4.05811075202487900426733149399, 5.61199802340024766492418490432, 6.93602322075311122737770532703, 7.82172125224434460852280240094, 8.751769741032712705264638958073, 9.385201676868890979143073732796, 10.31284878771040751335055391220, 10.94909120202510487410550012345, 12.47515486467872206818655866515

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