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Properties

Label 2-30e2-100.87-c1-0-34
Degree $2$
Conductor $900$
Sign $0.673 - 0.739i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + (0.221 + 1.39i)2-s + (−1.90 + 0.618i)4-s + (−1.59 − 1.56i)5-s + (−1.28 − 2.52i)8-s + (1.83 − 2.57i)10-s + (−0.552 + 3.48i)13-s + (3.23 − 2.35i)16-s + (6.76 − 3.44i)17-s + (3.99 + 2i)20-s + (0.0759 + 4.99i)25-s − 4.99·26-s + (9.93 − 3.22i)29-s + (4 + 4i)32-s + (6.30 + 8.68i)34-s + (11.3 + 1.80i)37-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (−0.712 − 0.701i)5-s + (−0.453 − 0.891i)8-s + (0.581 − 0.813i)10-s + (−0.153 + 0.967i)13-s + (0.809 − 0.587i)16-s + (1.63 − 0.835i)17-s + (0.894 + 0.447i)20-s + (0.0151 + 0.999i)25-s − 0.979·26-s + (1.84 − 0.599i)29-s + (0.707 + 0.707i)32-s + (1.08 + 1.48i)34-s + (1.87 + 0.296i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.673 - 0.739i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21462 + 0.536600i\)
\(L(\frac12)\) \(\approx\) \(1.21462 + 0.536600i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.221 - 1.39i)T \)
3 \( 1 \)
5 \( 1 + (1.59 + 1.56i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.552 - 3.48i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (-6.76 + 3.44i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (-9.93 + 3.22i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-11.3 - 1.80i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (-3.65 + 2.65i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (9.49 + 4.83i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.69 + 3.41i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-16.7 + 2.65i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (0.772 - 1.06i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.30 + 4.53i)T + (-57.0 - 78.4i)T^{2} \)
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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

−9.739964365036032578447599638808, −9.365189267930606324615071912800, −8.225591211049526552263186568190, −7.80071089881724944309685716910, −6.86725437969918229239670902586, −5.90252400536115918508772164036, −4.87100215809823389538985856091, −4.27948808292725656307937091892, −3.09714491439680765931503008400, −0.884599193708123836343392268660, 0.982439266663860697668567734168, 2.70652300459725008917553850014, 3.38776456217080705666779226319, 4.37891650874735823544159286855, 5.47371417433074418964786464113, 6.43134522180997960529976689439, 7.87199157452831733878337687569, 8.136706438272257431730504040197, 9.452402364262360356078397472080, 10.30002078451784276596417495104

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