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Properties

Label 2-30e2-20.3-c1-0-3
Degree $2$
Conductor $900$
Sign $-0.761 - 0.648i$
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  + (−1.40 − 0.178i)2-s + (1.93 + 0.5i)4-s + (−2.62 − 1.04i)8-s + 3.87i·11-s + (−2.44 − 2.44i)13-s + (3.50 + 1.93i)16-s + (−1.22 + 1.22i)17-s − 3.87·19-s + (0.690 − 5.43i)22-s + (−3.16 + 3.16i)23-s + (3 + 3.87i)26-s − 6i·29-s + 7.74i·31-s + (−4.56 − 3.34i)32-s + (1.93 − 1.5i)34-s + ⋯
L(s)  = 1  + (−0.992 − 0.126i)2-s + (0.968 + 0.250i)4-s + (−0.929 − 0.370i)8-s + 1.16i·11-s + (−0.679 − 0.679i)13-s + (0.875 + 0.484i)16-s + (−0.297 + 0.297i)17-s − 0.888·19-s + (0.147 − 1.15i)22-s + (−0.659 + 0.659i)23-s + (0.588 + 0.759i)26-s − 1.11i·29-s + 1.39i·31-s + (−0.807 − 0.590i)32-s + (0.332 − 0.257i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.761 - 0.648i$
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: \((2,\ 900,\ (\ :1/2),\ -0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115636 + 0.314065i\)
\(L(\frac12)\) \(\approx\) \(0.115636 + 0.314065i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.178i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 - 3.87iT - 11T^{2} \)
13 \( 1 + (2.44 + 2.44i)T + 13iT^{2} \)
17 \( 1 + (1.22 - 1.22i)T - 17iT^{2} \)
19 \( 1 + 3.87T + 19T^{2} \)
23 \( 1 + (3.16 - 3.16i)T - 23iT^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 7.74iT - 31T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 37iT^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (3.16 + 3.16i)T + 47iT^{2} \)
53 \( 1 + (-2.44 - 2.44i)T + 53iT^{2} \)
59 \( 1 + 7.74T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + (-4.74 - 4.74i)T + 67iT^{2} \)
71 \( 1 - 7.74iT - 71T^{2} \)
73 \( 1 + (-3.67 - 3.67i)T + 73iT^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + (1.58 - 1.58i)T - 83iT^{2} \)
89 \( 1 - 9iT - 89T^{2} \)
97 \( 1 + (4.89 - 4.89i)T - 97iT^{2} \)
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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

−10.15737717947072501955092584347, −9.809032536192154851138598643914, −8.742199775507361866625594342078, −8.000064622898099373166457801219, −7.19268370673233410061130161041, −6.43748085108234934104287572965, −5.27493817931315304957449747895, −4.04111160918184999482733286472, −2.69882692030268898499049648299, −1.68345689602823272444281369206, 0.20594695195607747497155001270, 1.89780235075696493994856525734, 3.00819027609171714614910656627, 4.39759711828688116244447466776, 5.73484489359246758974023830518, 6.45833453378184931441192071083, 7.35357339615788529437759813640, 8.231385817821510584498901015850, 8.962107119592271295073977062950, 9.611666694361313251241248001571

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