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Properties

Label 2-30e2-20.7-c1-0-8
Degree $2$
Conductor $900$
Sign $-0.828 - 0.559i$
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  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−3.15 + 3.15i)7-s + 2.82i·8-s − 2i·11-s + (−1.60 + 1.60i)13-s + (−6.09 + 1.63i)14-s + (−2.00 + 3.46i)16-s + (4.24 + 4.24i)17-s − 3.19·19-s + (1.41 − 2.44i)22-s + (−5.27 − 5.27i)23-s + (−3.09 + 0.830i)26-s + (−8.62 − 2.31i)28-s − 0.535i·29-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.499 + 0.866i)4-s + (−1.19 + 1.19i)7-s + 0.999i·8-s − 0.603i·11-s + (−0.444 + 0.444i)13-s + (−1.62 + 0.436i)14-s + (−0.500 + 0.866i)16-s + (1.02 + 1.02i)17-s − 0.733·19-s + (0.301 − 0.522i)22-s + (−1.10 − 1.10i)23-s + (−0.607 + 0.162i)26-s + (−1.62 − 0.436i)28-s − 0.0995i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.828 - 0.559i$
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: \((2,\ 900,\ (\ :1/2),\ -0.828 - 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.538026 + 1.75700i\)
\(L(\frac12)\) \(\approx\) \(0.538026 + 1.75700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (3.15 - 3.15i)T - 7iT^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (1.60 - 1.60i)T - 13iT^{2} \)
17 \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \)
19 \( 1 + 3.19T + 19T^{2} \)
23 \( 1 + (5.27 + 5.27i)T + 23iT^{2} \)
29 \( 1 + 0.535iT - 29T^{2} \)
31 \( 1 - 3.73iT - 31T^{2} \)
37 \( 1 + (-7.72 - 7.72i)T + 37iT^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 + (-3.15 - 3.15i)T + 43iT^{2} \)
47 \( 1 + (0.656 - 0.656i)T - 47iT^{2} \)
53 \( 1 + (3.86 - 3.86i)T - 53iT^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + (-3.91 + 3.91i)T - 67iT^{2} \)
71 \( 1 - 2.53iT - 71T^{2} \)
73 \( 1 + (-2.82 + 2.82i)T - 73iT^{2} \)
79 \( 1 - 7.46T + 79T^{2} \)
83 \( 1 + (-9.14 - 9.14i)T + 83iT^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + (-6.88 - 6.88i)T + 97iT^{2} \)
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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

−10.45335408235620663809363179755, −9.550049902935615389540156134287, −8.537693939443136364157652399496, −7.987441910174797115957782248549, −6.53935912626926040717682640344, −6.23887868729350800968986036631, −5.38375819260324672280694444978, −4.18411493053038748698204779333, −3.18968454398108628724799937484, −2.29156355337938149730520093586, 0.62405684757798701195565561912, 2.32393355316460838986935809704, 3.48847857371927639188247029619, 4.13358358507043527774673227393, 5.28768564581241677780595783238, 6.19700865549402119881085385371, 7.14009751664855658723941245722, 7.69666725309882282710198975054, 9.547019333520655316911227720965, 9.818783202428854878448453658687

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