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Properties

Label 2-30e2-20.3-c1-0-35
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  + (0.707 + 1.22i)2-s + (−0.999 + 1.73i)4-s + (−1.74 − 1.74i)7-s − 2.82·8-s − 2i·11-s + (−4.05 − 4.05i)13-s + (0.901 − 3.36i)14-s + (−2.00 − 3.46i)16-s + (4.24 − 4.24i)17-s − 7.19·19-s + (2.44 − 1.41i)22-s + (0.378 − 0.378i)23-s + (2.09 − 7.83i)26-s + (4.76 − 1.27i)28-s + 7.46i·29-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.658 − 0.658i)7-s − 0.999·8-s − 0.603i·11-s + (−1.12 − 1.12i)13-s + (0.241 − 0.899i)14-s + (−0.500 − 0.866i)16-s + (1.02 − 1.02i)17-s − 1.65·19-s + (0.522 − 0.301i)22-s + (0.0790 − 0.0790i)23-s + (0.411 − 1.53i)26-s + (0.899 − 0.241i)28-s + 1.38i·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

\(L(1)\) \(\approx\) \(0.666891 - 0.445484i\)
\(L(\frac12)\) \(\approx\) \(0.666891 - 0.445484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (1.74 + 1.74i)T + 7iT^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (4.05 + 4.05i)T + 13iT^{2} \)
17 \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \)
19 \( 1 + 7.19T + 19T^{2} \)
23 \( 1 + (-0.378 + 0.378i)T - 23iT^{2} \)
29 \( 1 - 7.46iT - 29T^{2} \)
31 \( 1 - 0.267iT - 31T^{2} \)
37 \( 1 + (2.07 - 2.07i)T - 37iT^{2} \)
41 \( 1 - 5.46T + 41T^{2} \)
43 \( 1 + (-1.74 + 1.74i)T - 43iT^{2} \)
47 \( 1 + (9.14 + 9.14i)T + 47iT^{2} \)
53 \( 1 + (-1.03 - 1.03i)T + 53iT^{2} \)
59 \( 1 + 4.53T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + (8.81 + 8.81i)T + 67iT^{2} \)
71 \( 1 - 9.46iT - 71T^{2} \)
73 \( 1 + (-2.82 - 2.82i)T + 73iT^{2} \)
79 \( 1 + 0.535T + 79T^{2} \)
83 \( 1 + (-0.656 + 0.656i)T - 83iT^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + (-4.43 + 4.43i)T - 97iT^{2} \)
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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

−9.930314965815902760103147170680, −8.946248531354711210633359817637, −8.062788019629364061286224882113, −7.27682714763513460939734257924, −6.59323748525358500833055167105, −5.55692038319577513097241733095, −4.81359579044477279109991368009, −3.60497601275417520229077323474, −2.82860492725450950558732359544, −0.30564100135071328043156823593, 1.84975702889901780813187639998, 2.68670950710936571775463315334, 3.99264179241068921200360924740, 4.69835259776088176052739560868, 5.92195608341675229532188961255, 6.49883621846037809272282556421, 7.80726041422429517995635427715, 8.952456794623282951644398262756, 9.615766035395167151615997014269, 10.21217554279769208772416755913

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