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Properties

Label 2-15e2-225.106-c1-0-27
Degree $2$
Conductor $225$
Sign $-0.913 - 0.406i$
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  + (0.322 + 0.0684i)2-s + (−0.519 − 1.65i)3-s + (−1.72 − 0.769i)4-s + (−1.71 + 1.42i)5-s + (−0.0543 − 0.567i)6-s + (−1.22 + 2.11i)7-s + (−1.03 − 0.753i)8-s + (−2.45 + 1.71i)9-s + (−0.651 + 0.342i)10-s + (−1.65 − 0.351i)11-s + (−0.372 + 3.25i)12-s + (2.11 − 0.450i)13-s + (−0.537 + 0.597i)14-s + (3.25 + 2.09i)15-s + (2.24 + 2.49i)16-s + (−3.09 − 2.24i)17-s + ⋯
L(s)  = 1  + (0.227 + 0.0484i)2-s + (−0.300 − 0.953i)3-s + (−0.863 − 0.384i)4-s + (−0.769 + 0.639i)5-s + (−0.0221 − 0.231i)6-s + (−0.461 + 0.798i)7-s + (−0.366 − 0.266i)8-s + (−0.819 + 0.572i)9-s + (−0.206 + 0.108i)10-s + (−0.498 − 0.106i)11-s + (−0.107 + 0.939i)12-s + (0.587 − 0.124i)13-s + (−0.143 + 0.159i)14-s + (0.840 + 0.541i)15-s + (0.562 + 0.624i)16-s + (−0.750 − 0.545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.913 - 0.406i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.913 - 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00723819 + 0.0340541i\)
\(L(\frac12)\) \(\approx\) \(0.00723819 + 0.0340541i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.519 + 1.65i)T \)
5 \( 1 + (1.71 - 1.42i)T \)
good2 \( 1 + (-0.322 - 0.0684i)T + (1.82 + 0.813i)T^{2} \)
7 \( 1 + (1.22 - 2.11i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.65 + 0.351i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-2.11 + 0.450i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (3.09 + 2.24i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.15 + 3.01i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (5.83 - 6.48i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (0.606 + 5.76i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (-0.743 + 7.07i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-2.56 + 7.88i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (4.23 - 0.900i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (3.28 - 5.69i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.832 - 7.92i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (2.04 - 1.48i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.942 + 0.200i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-8.52 - 1.81i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (0.353 - 3.36i)T + (-65.5 - 13.9i)T^{2} \)
71 \( 1 + (0.529 - 0.384i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.37 - 10.3i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.658 + 6.26i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (13.0 - 5.81i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (2.96 + 9.11i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-0.161 - 1.53i)T + (-94.8 + 20.1i)T^{2} \)
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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

−11.71386910469071921965016540066, −11.04730983545102213302974790598, −9.700302107645781328165228632617, −8.571680281173967475072687414348, −7.70156915154385513654956787058, −6.38112585296164464573761586176, −5.67705909520173703996436800310, −4.15427304783668529918640171791, −2.57624268325021739774646053411, −0.02727654911378826868028215214, 3.56412046273451583336705373170, 4.20143839237050562034486516178, 5.11516728590048004766846801006, 6.58893993766464976630098380554, 8.359838783797533025119918151225, 8.634141537642971402051043725303, 10.05010466509436454773205705343, 10.69881575710821196727849136445, 11.98574991713354444905242840578, 12.70497403521815089436355339220

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