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Properties

Label 2-30e2-20.7-c1-0-6
Degree $2$
Conductor $900$
Sign $0.980 - 0.198i$
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 + 3.87i·11-s + (−2.44 + 2.44i)13-s + (3.50 + 1.93i)16-s + (−1.22 − 1.22i)17-s + 3.87·19-s + (5.43 + 0.690i)22-s + (3.16 + 3.16i)23-s + (3 + 3.87i)26-s + 6i·29-s + 7.74i·31-s + (3.34 − 4.56i)32-s + (−1.93 + 1.5i)34-s + ⋯
L(s)  = 1  + (0.126 − 0.992i)2-s + (−0.968 − 0.250i)4-s + (−0.370 + 0.929i)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 + (1.15 + 0.147i)22-s + (0.659 + 0.659i)23-s + (0.588 + 0.759i)26-s + 1.11i·29-s + 1.39i·31-s + (0.590 − 0.807i)32-s + (−0.332 + 0.257i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.980 - 0.198i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.980 - 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18115 + 0.118639i\)
\(L(\frac12)\) \(\approx\) \(1.18115 + 0.118639i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.87iT - 11T^{2} \)
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

−10.19996580502316537552263953222, −9.351469209880856370074508481299, −8.915248350998640908119042073551, −7.54832940893396833059714959422, −6.86823207211743431626237070662, −5.33389028404346933741049403598, −4.77886921495879807831867889866, −3.68208746017667502944352299687, −2.55887910859226055529409331781, −1.44349631442565985711184635705, 0.58470639124225478767418693744, 2.81876732814800334472885214702, 3.89483823326574150972669125477, 5.00707040306826579663215045364, 5.78837212375006985952291180393, 6.58182970669288541085761844310, 7.62421440301509828480385506115, 8.206207276685805843610743237752, 9.091209135415821445436129125625, 9.872919682002470531368800505994

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