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Properties

Label 2-30e2-4.3-c0-0-1
Degree $2$
Conductor $900$
Sign $1$
Analytic cond. $0.449158$
Root an. cond. $0.670192$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 16-s − 2·17-s + 32-s − 2·34-s + 49-s − 2·53-s − 2·61-s + 64-s − 2·68-s + 98-s − 2·106-s − 2·109-s + 2·113-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s + 16-s − 2·17-s + 32-s − 2·34-s + 49-s − 2·53-s − 2·61-s + 64-s − 2·68-s + 98-s − 2·106-s − 2·109-s + 2·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.784580502\)
\(L(\frac12)\) \(\approx\) \(1.784580502\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.68682024255314414976064779512, −9.527873296643808157033084528787, −8.580970433215067554859337494509, −7.55066885983855843486372146244, −6.69342016108698868656279910809, −5.99673558689662626326706314773, −4.85831092386239839105305591669, −4.18064729672705212396731265232, −2.98395527244364635313969528176, −1.89925069900852131232230684548, 1.89925069900852131232230684548, 2.98395527244364635313969528176, 4.18064729672705212396731265232, 4.85831092386239839105305591669, 5.99673558689662626326706314773, 6.69342016108698868656279910809, 7.55066885983855843486372146244, 8.580970433215067554859337494509, 9.527873296643808157033084528787, 10.68682024255314414976064779512

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