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Properties

Label 2-30e2-20.3-c1-0-13
Degree $2$
Conductor $900$
Sign $0.956 + 0.290i$
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  + (0.581 − 1.28i)2-s + (−1.32 − 1.50i)4-s + (1.41 + 1.41i)7-s + (−2.70 + 0.832i)8-s + 5.29i·11-s + (3.74 + 3.74i)13-s + (2.64 − 1.00i)14-s + (−0.5 + 3.96i)16-s + 5.29·19-s + (6.82 + 3.07i)22-s + (−2.82 + 2.82i)23-s + (7 − 2.64i)26-s + (0.250 − 3.99i)28-s − 8i·29-s − 5.29i·31-s + (4.82 + 2.95i)32-s + ⋯
L(s)  = 1  + (0.411 − 0.911i)2-s + (−0.661 − 0.750i)4-s + (0.534 + 0.534i)7-s + (−0.955 + 0.294i)8-s + 1.59i·11-s + (1.03 + 1.03i)13-s + (0.707 − 0.267i)14-s + (−0.125 + 0.992i)16-s + 1.21·19-s + (1.45 + 0.656i)22-s + (−0.589 + 0.589i)23-s + (1.37 − 0.518i)26-s + (0.0473 − 0.754i)28-s − 1.48i·29-s − 0.950i·31-s + (0.852 + 0.522i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.956 + 0.290i$
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: \((2,\ 900,\ (\ :1/2),\ 0.956 + 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85052 - 0.274465i\)
\(L(\frac12)\) \(\approx\) \(1.85052 - 0.274465i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.581 + 1.28i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1.41 - 1.41i)T + 7iT^{2} \)
11 \( 1 - 5.29iT - 11T^{2} \)
13 \( 1 + (-3.74 - 3.74i)T + 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 + 5.29iT - 31T^{2} \)
37 \( 1 + (3.74 - 3.74i)T - 37iT^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + (5.65 - 5.65i)T - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-7.48 - 7.48i)T + 53iT^{2} \)
59 \( 1 - 5.29T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (8.48 + 8.48i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-7.48 - 7.48i)T + 73iT^{2} \)
79 \( 1 + 5.29T + 79T^{2} \)
83 \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (7.48 - 7.48i)T - 97iT^{2} \)
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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

−9.923630462162784000154830754459, −9.577348226750832765568961659062, −8.611764180082469800977881568117, −7.61472441015577319248262275950, −6.43224749097824072838401633708, −5.47952322343352481463135088687, −4.55782677831363412819827913605, −3.78761770485133223033312809650, −2.35137112726245233116741889310, −1.51085356921619075951577113079, 0.885353228704259719854496894554, 3.18728094092266092242274587010, 3.76564659472942343574422059546, 5.17163904276694048414565334965, 5.68251332195997280680479773702, 6.69590852971887560075678551893, 7.59678634757795049537863785114, 8.459287441216007726400849686240, 8.806091391681985188456611215817, 10.22990079776072333786853115333

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