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Properties

Label 2-30e2-20.3-c1-0-1
Degree $2$
Conductor $900$
Sign $-0.692 + 0.721i$
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  + (0.178 + 1.40i)2-s + (−1.93 + 0.5i)4-s + (−1.04 − 2.62i)8-s + (3.50 − 1.93i)16-s + (−4.89 + 4.89i)17-s − 7.74·19-s + (−6.32 + 6.32i)23-s − 7.74i·31-s + (3.34 + 4.56i)32-s + (−7.74 − 6i)34-s + (−1.38 − 10.8i)38-s + (−10.0 − 7.74i)46-s + (−6.32 − 6.32i)47-s − 7i·49-s + (9.79 + 9.79i)53-s + ⋯
L(s)  = 1  + (0.126 + 0.992i)2-s + (−0.968 + 0.250i)4-s + (−0.370 − 0.929i)8-s + (0.875 − 0.484i)16-s + (−1.18 + 1.18i)17-s − 1.77·19-s + (−1.31 + 1.31i)23-s − 1.39i·31-s + (0.590 + 0.807i)32-s + (−1.32 − 1.02i)34-s + (−0.223 − 1.76i)38-s + (−1.47 − 1.14i)46-s + (−0.922 − 0.922i)47-s − i·49-s + (1.34 + 1.34i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.692 + 0.721i$
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: \((2,\ 900,\ (\ :1/2),\ -0.692 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.139337 - 0.326711i\)
\(L(\frac12)\) \(\approx\) \(0.139337 - 0.326711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.178 - 1.40i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (4.89 - 4.89i)T - 17iT^{2} \)
19 \( 1 + 7.74T + 19T^{2} \)
23 \( 1 + (6.32 - 6.32i)T - 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 7.74iT - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (6.32 + 6.32i)T + 47iT^{2} \)
53 \( 1 + (-9.79 - 9.79i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 7.74T + 79T^{2} \)
83 \( 1 + (12.6 - 12.6i)T - 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−10.44808867247185886252074382399, −9.666099471865204497414740256867, −8.648055984815172821245172437538, −8.192590087105717193629333019550, −7.16384847604314617632588006606, −6.28119354352638834374089664609, −5.69074031600010374641709025177, −4.36743366559801587060041281234, −3.83346014350355159343629248541, −2.06246969767155114325294797495, 0.15316117679377858447187782426, 1.95006146493499595497982547531, 2.85366723689828712933563529376, 4.21100166742773665026037194135, 4.74716963183072940015233090873, 6.02552247219850327584622659927, 6.91414499323989511749952096451, 8.328293266930894643507258380122, 8.770344855222277239902622117431, 9.763267522244841967459107795932

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