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Properties

Label 2-30e2-25.9-c1-0-4
Degree $2$
Conductor $900$
Sign $0.740 - 0.672i$
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  + (1.64 − 1.51i)5-s + 3.78i·7-s + (−0.653 − 0.474i)11-s + (2.79 + 3.84i)13-s + (1.09 + 0.355i)17-s + (−0.00463 + 0.0142i)19-s + (−3.68 + 5.07i)23-s + (0.395 − 4.98i)25-s + (1.14 + 3.51i)29-s + (−0.488 + 1.50i)31-s + (5.74 + 6.22i)35-s + (5.02 + 6.91i)37-s + (9.30 − 6.75i)41-s − 10.2i·43-s + (−0.500 + 0.162i)47-s + ⋯
L(s)  = 1  + (0.734 − 0.678i)5-s + 1.43i·7-s + (−0.197 − 0.143i)11-s + (0.774 + 1.06i)13-s + (0.264 + 0.0861i)17-s + (−0.00106 + 0.00327i)19-s + (−0.768 + 1.05i)23-s + (0.0790 − 0.996i)25-s + (0.212 + 0.653i)29-s + (−0.0878 + 0.270i)31-s + (0.971 + 1.05i)35-s + (0.825 + 1.13i)37-s + (1.45 − 1.05i)41-s − 1.56i·43-s + (−0.0730 + 0.0237i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.740 - 0.672i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.740 - 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65136 + 0.638271i\)
\(L(\frac12)\) \(\approx\) \(1.65136 + 0.638271i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 1.51i)T \)
good7 \( 1 - 3.78iT - 7T^{2} \)
11 \( 1 + (0.653 + 0.474i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.79 - 3.84i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.09 - 0.355i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.00463 - 0.0142i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.68 - 5.07i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.14 - 3.51i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.488 - 1.50i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.02 - 6.91i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.30 + 6.75i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + (0.500 - 0.162i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.80 + 0.911i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.25 - 6.72i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.54 + 1.84i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-12.6 - 4.10i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.51 - 4.67i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.75 + 3.78i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.86 + 8.81i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.35 - 0.439i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-13.0 - 9.46i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (7.66 - 2.49i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.03379689309076076466654077080, −9.122777050978886253879643270874, −8.848600113344077553048661712185, −7.87080728206626273854599289549, −6.52461447126422387702317689488, −5.78351614643570547273039463740, −5.16212794518147497645959721811, −3.92295726201821841564673856036, −2.51308241902518130359797599972, −1.53097114569634920375244628327, 0.926379351814646784793353438405, 2.50055602943816883988681813683, 3.59606016523121420054889408326, 4.57804657151857878347269090406, 5.89642380026973258470102452738, 6.46073055466228973030287341139, 7.57805856348957062842627778167, 8.053475860863066644449803749548, 9.465738269119380295323329012726, 10.09541383171701109754063476358

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