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Properties

Label 2-30e2-20.19-c2-0-83
Degree $2$
Conductor $900$
Sign $-0.951 - 0.308i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.758 − 1.85i)2-s + (−2.85 − 2.80i)4-s + 4.09·7-s + (−7.35 + 3.14i)8-s − 0.984i·11-s − 14.8i·13-s + (3.10 − 7.58i)14-s + (0.253 + 15.9i)16-s + 9.80i·17-s − 27.3i·19-s + (−1.82 − 0.746i)22-s − 18.3·23-s + (−27.4 − 11.2i)26-s + (−11.6 − 11.4i)28-s − 2.80·29-s + ⋯
L(s)  = 1  + (0.379 − 0.925i)2-s + (−0.712 − 0.701i)4-s + 0.585·7-s + (−0.919 + 0.393i)8-s − 0.0894i·11-s − 1.13i·13-s + (0.221 − 0.541i)14-s + (0.0158 + 0.999i)16-s + 0.576i·17-s − 1.44i·19-s + (−0.0828 − 0.0339i)22-s − 0.797·23-s + (−1.05 − 0.431i)26-s + (−0.417 − 0.410i)28-s − 0.0967·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.951 - 0.308i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.951 - 0.308i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.261367669\)
\(L(\frac12)\) \(\approx\) \(1.261367669\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.758 + 1.85i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4.09T + 49T^{2} \)
11 \( 1 + 0.984iT - 121T^{2} \)
13 \( 1 + 14.8iT - 169T^{2} \)
17 \( 1 - 9.80iT - 289T^{2} \)
19 \( 1 + 27.3iT - 361T^{2} \)
23 \( 1 + 18.3T + 529T^{2} \)
29 \( 1 + 2.80T + 841T^{2} \)
31 \( 1 + 1.96iT - 961T^{2} \)
37 \( 1 + 44.8iT - 1.36e3T^{2} \)
41 \( 1 + 17T + 1.68e3T^{2} \)
43 \( 1 + 54.8T + 1.84e3T^{2} \)
47 \( 1 + 58.8T + 2.20e3T^{2} \)
53 \( 1 + 89.2iT - 2.80e3T^{2} \)
59 \( 1 - 67.3iT - 3.48e3T^{2} \)
61 \( 1 - 36.0T + 3.72e3T^{2} \)
67 \( 1 - 25.5T + 4.48e3T^{2} \)
71 \( 1 - 120. iT - 5.04e3T^{2} \)
73 \( 1 + 69.8iT - 5.32e3T^{2} \)
79 \( 1 - 32.2iT - 6.24e3T^{2} \)
83 \( 1 + 68.1T + 6.88e3T^{2} \)
89 \( 1 + 67.8T + 7.92e3T^{2} \)
97 \( 1 - 1.16iT - 9.40e3T^{2} \)
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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

−9.721696928891234291672675360662, −8.647123151525141592957034785303, −8.051679824076224872347903235241, −6.75018716853330601599298672220, −5.62183474203398007520206478101, −4.94369690345869891704006790121, −3.89677021281771341282494515375, −2.87217003215777695404485589388, −1.74482837855546867779378725189, −0.35344087511304212339605688959, 1.72744164792531293253973251330, 3.33599258847794613250205515497, 4.35475331465812495232570854791, 5.09897043625122041864173103361, 6.15238722788989844494857037791, 6.86501610576924377020964269943, 7.88020457259698685794513980091, 8.395178927794824263794314481836, 9.428184030895433434245724437669, 10.12043874044769871772831436430

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