Properties

Label 2-30e2-1.1-c1-0-7
Degree $2$
Conductor $900$
Sign $-1$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 6·11-s − 5·13-s − 6·17-s + 5·19-s − 6·23-s + 6·29-s − 31-s − 2·37-s + 43-s + 6·47-s − 6·49-s − 12·53-s + 6·59-s − 13·61-s − 11·67-s − 2·73-s − 6·77-s + 8·79-s − 6·83-s − 5·91-s + 7·97-s + 12·101-s + 4·103-s + 12·107-s − 7·109-s + 12·113-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.80·11-s − 1.38·13-s − 1.45·17-s + 1.14·19-s − 1.25·23-s + 1.11·29-s − 0.179·31-s − 0.328·37-s + 0.152·43-s + 0.875·47-s − 6/7·49-s − 1.64·53-s + 0.781·59-s − 1.66·61-s − 1.34·67-s − 0.234·73-s − 0.683·77-s + 0.900·79-s − 0.658·83-s − 0.524·91-s + 0.710·97-s + 1.19·101-s + 0.394·103-s + 1.16·107-s − 0.670·109-s + 1.12·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−9.853259570123841667190565291355, −8.821114032953874975275414885204, −7.82946935002197249360095495788, −7.40049250827422786308554046568, −6.17945227400387495932931920930, −5.09151237655519270484018689309, −4.57579314482224407000052258764, −2.97401840613225752286969945588, −2.10805618754928013340727410523, 0, 2.10805618754928013340727410523, 2.97401840613225752286969945588, 4.57579314482224407000052258764, 5.09151237655519270484018689309, 6.17945227400387495932931920930, 7.40049250827422786308554046568, 7.82946935002197249360095495788, 8.821114032953874975275414885204, 9.853259570123841667190565291355

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