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Properties

Label 2-30e2-25.16-c1-0-1
Degree $2$
Conductor $900$
Sign $-0.387 - 0.921i$
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  + (−2.18 − 0.464i)5-s + 0.547·7-s + (−1.08 − 0.786i)11-s + (−0.244 + 0.177i)13-s + (−1.24 + 3.84i)17-s + (−1.74 + 5.35i)19-s + (0.198 + 0.144i)23-s + (4.56 + 2.03i)25-s + (0.423 + 1.30i)29-s + (−1.09 + 3.36i)31-s + (−1.19 − 0.254i)35-s + (1.76 − 1.28i)37-s + (−7.93 + 5.76i)41-s − 8.35·43-s + (3.23 + 9.96i)47-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)5-s + 0.206·7-s + (−0.326 − 0.237i)11-s + (−0.0677 + 0.0492i)13-s + (−0.302 + 0.932i)17-s + (−0.399 + 1.22i)19-s + (0.0413 + 0.0300i)23-s + (0.913 + 0.406i)25-s + (0.0785 + 0.241i)29-s + (−0.196 + 0.604i)31-s + (−0.202 − 0.0430i)35-s + (0.290 − 0.211i)37-s + (−1.23 + 0.899i)41-s − 1.27·43-s + (0.472 + 1.45i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.387 - 0.921i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.387 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.380925 + 0.573338i\)
\(L(\frac12)\) \(\approx\) \(0.380925 + 0.573338i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 + 0.464i)T \)
good7 \( 1 - 0.547T + 7T^{2} \)
11 \( 1 + (1.08 + 0.786i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.244 - 0.177i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.24 - 3.84i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.74 - 5.35i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.198 - 0.144i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.423 - 1.30i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.09 - 3.36i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.76 + 1.28i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.93 - 5.76i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.35T + 43T^{2} \)
47 \( 1 + (-3.23 - 9.96i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.37 - 7.31i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.35 + 2.44i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.67 + 1.22i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.62 + 8.07i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.83 - 8.73i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.86 + 6.43i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.69 - 14.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.05 + 6.32i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-4.02 - 2.92i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.25 - 3.86i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.54277659029885449057695412933, −9.522612366855251078967458300498, −8.318999298502398285070903076179, −8.162638509898549227171325578503, −7.03671295239996180381307210148, −6.08096861417609699993388910315, −4.98090225440878666108496513951, −4.07239056851943545496887441537, −3.15398149708250122784788597372, −1.54681489572170553379701142582, 0.32984632297464674091433006358, 2.30873275946145849607624970696, 3.41854860129504245659114487001, 4.52384077143697673615520269628, 5.23751307260214430586275366837, 6.70378753951683868067275947093, 7.22099841645688667830665976664, 8.186216539178594961182935753855, 8.870340137563425297201827727322, 9.910924151320027500389371684136

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