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Properties

Label 2-330e2-1.1-c1-0-10
Degree $2$
Conductor $108900$
Sign $1$
Analytic cond. $869.570$
Root an. cond. $29.4884$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 4·13-s + 4·19-s − 6·23-s − 6·29-s + 8·31-s − 2·37-s + 6·41-s + 8·43-s + 6·47-s + 9·49-s − 6·53-s + 12·59-s − 2·61-s + 10·67-s + 12·71-s − 16·73-s − 8·79-s − 6·89-s + 16·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.10·13-s + 0.917·19-s − 1.25·23-s − 1.11·29-s + 1.43·31-s − 0.328·37-s + 0.937·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s − 0.824·53-s + 1.56·59-s − 0.256·61-s + 1.22·67-s + 1.42·71-s − 1.87·73-s − 0.900·79-s − 0.635·89-s + 1.67·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(869.570\)
Root analytic conductor: \(29.4884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 108900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9717379088\)
\(L(\frac12)\) \(\approx\) \(0.9717379088\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.74574568709664, −13.10968001204094, −12.60017023235336, −12.34505198075580, −11.83792061370499, −11.27723369702003, −10.67678360488925, −10.00877841023716, −9.657190172665080, −9.572513431483686, −8.782750148128602, −8.200827808311086, −7.515686055960293, −7.234137772902222, −6.635653592907186, −6.064482989757100, −5.628870579397701, −5.101093011636895, −4.167773930986533, −3.965635773098285, −3.116459236005854, −2.668175266588378, −2.149296579155488, −1.112261815572504, −0.3298052217283841, 0.3298052217283841, 1.112261815572504, 2.149296579155488, 2.668175266588378, 3.116459236005854, 3.965635773098285, 4.167773930986533, 5.101093011636895, 5.628870579397701, 6.064482989757100, 6.635653592907186, 7.234137772902222, 7.515686055960293, 8.200827808311086, 8.782750148128602, 9.572513431483686, 9.657190172665080, 10.00877841023716, 10.67678360488925, 11.27723369702003, 11.83792061370499, 12.34505198075580, 12.60017023235336, 13.10968001204094, 13.74574568709664

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