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Properties

Label 2-30e2-20.7-c1-0-32
Degree $2$
Conductor $900$
Sign $0.0113 + 0.999i$
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  + (0.394 + 1.35i)2-s + (−1.68 + 1.07i)4-s + (−2.47 + 2.47i)7-s + (−2.11 − 1.87i)8-s − 3.02i·11-s + (−0.363 + 0.363i)13-s + (−4.34 − 2.38i)14-s + (1.70 − 3.61i)16-s + (−2.36 − 2.36i)17-s − 4.95·19-s + (4.11 − 1.19i)22-s + (−0.900 − 0.900i)23-s + (−0.636 − 0.350i)26-s + (1.53 − 6.83i)28-s − 3.50i·29-s + ⋯
L(s)  = 1  + (0.278 + 0.960i)2-s + (−0.844 + 0.535i)4-s + (−0.936 + 0.936i)7-s + (−0.749 − 0.661i)8-s − 0.913i·11-s + (−0.100 + 0.100i)13-s + (−1.16 − 0.638i)14-s + (0.426 − 0.904i)16-s + (−0.573 − 0.573i)17-s − 1.13·19-s + (0.876 − 0.254i)22-s + (−0.187 − 0.187i)23-s + (−0.124 − 0.0686i)26-s + (0.289 − 1.29i)28-s − 0.650i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0113 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0113 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0113 + 0.999i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.0113 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0599475 - 0.0592722i\)
\(L(\frac12)\) \(\approx\) \(0.0599475 - 0.0592722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.394 - 1.35i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (2.47 - 2.47i)T - 7iT^{2} \)
11 \( 1 + 3.02iT - 11T^{2} \)
13 \( 1 + (0.363 - 0.363i)T - 13iT^{2} \)
17 \( 1 + (2.36 + 2.36i)T + 17iT^{2} \)
19 \( 1 + 4.95T + 19T^{2} \)
23 \( 1 + (0.900 + 0.900i)T + 23iT^{2} \)
29 \( 1 + 3.50iT - 29T^{2} \)
31 \( 1 - 3.85iT - 31T^{2} \)
37 \( 1 + (-0.363 - 0.363i)T + 37iT^{2} \)
41 \( 1 + 2.72T + 41T^{2} \)
43 \( 1 + (3.92 + 3.92i)T + 43iT^{2} \)
47 \( 1 + (-5.85 + 5.85i)T - 47iT^{2} \)
53 \( 1 + (-3.14 + 3.14i)T - 53iT^{2} \)
59 \( 1 + 8.68T + 59T^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 + (-3.92 + 3.92i)T - 67iT^{2} \)
71 \( 1 - 4.25iT - 71T^{2} \)
73 \( 1 + (9.28 - 9.28i)T - 73iT^{2} \)
79 \( 1 - 0.399T + 79T^{2} \)
83 \( 1 + (0.199 + 0.199i)T + 83iT^{2} \)
89 \( 1 + 4.28iT - 89T^{2} \)
97 \( 1 + (6.73 + 6.73i)T + 97iT^{2} \)
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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

−9.598079919684013019022013087534, −8.831435649695084302700609608992, −8.374212003230965572899796118545, −7.14026163620038142093260818979, −6.34250709170920146323284796538, −5.78004083878989952003841917609, −4.73063330431620849575037297302, −3.59836304247401092168481201874, −2.58507513257876334400968871948, −0.03521191405069746983558634235, 1.67899890750585748949958102876, 2.92436974319767521524220335200, 4.02518960472282770727285251274, 4.59785646885571716386938534769, 5.96809467769493928014026749539, 6.77684194577450095961638855390, 7.84822848138842212086172249843, 8.985606028361989014599646903590, 9.675762113371709906265016735164, 10.47734062975500452442500129469

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