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Properties

Label 2-30e2-20.3-c1-0-10
Degree $2$
Conductor $900$
Sign $-0.382 - 0.923i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.579568 + 0.867615i\)
\(L(\frac12)\) \(\approx\) \(0.579568 + 0.867615i\)
\(L(\frac{3}{2})\) 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

−10.38268353652655281080641644987, −9.228871241084918929351801013214, −8.541055460075168975107347976003, −8.240095248077177564029349916082, −6.93420015023120436535368373217, −6.27592334238752580110035886555, −5.22152636237670956760197834988, −4.43483190047360802907600168418, −2.41921483946898846923152881697, −1.62827877062494839081592894571, 0.68068694918993886199676050362, 1.87899933841634962951419531677, 3.29878623752087529210903791581, 4.25378462640446652994271272273, 5.36648297373030779516388799726, 6.93825113781018716798373344153, 7.30569809769691852559398805998, 8.433814948442020429116256917004, 8.813325947523001255256811665122, 9.996583010095278735978309120116

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