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Properties

Label 2-30e2-5.4-c1-0-3
Degree $2$
Conductor $900$
Sign $0.894 - 0.447i$
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  − i·7-s + 7i·13-s + 7·19-s + 11·31-s − 10i·37-s + 13i·43-s + 6·49-s − 61-s + 11i·67-s + 10i·73-s + 4·79-s + 7·91-s − 19i·97-s − 20i·103-s − 17·109-s + ⋯
L(s)  = 1  − 0.377i·7-s + 1.94i·13-s + 1.60·19-s + 1.97·31-s − 1.64i·37-s + 1.98i·43-s + 0.857·49-s − 0.128·61-s + 1.34i·67-s + 1.17i·73-s + 0.450·79-s + 0.733·91-s − 1.92i·97-s − 1.97i·103-s − 1.62·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55156 + 0.366275i\)
\(L(\frac12)\) \(\approx\) \(1.55156 + 0.366275i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 7iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 11T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 11iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 19iT - 97T^{2} \)
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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

−9.977823893178558580778723791723, −9.450385650233187244685527500150, −8.564714861902500639015066645338, −7.50897121623622279840396021121, −6.84483149746988145169386310723, −5.90331239581687517956669189722, −4.72051500497346810470198581000, −3.96035182333274418898081674294, −2.66349277684815050575823334823, −1.26319395930967265118682164881, 0.925200601601659279689990278024, 2.67339752721048925628385533629, 3.46402868061878143943868317573, 4.94728532785869565693476466944, 5.57020635769809955483359577206, 6.57066381528960798485575627732, 7.71144063614632070805021969707, 8.219346884431257290134254543856, 9.261099106086246421849458954556, 10.14112622647243967370640998702

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