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Properties

Label 2-385e2-1.1-c1-0-47
Degree $2$
Conductor $148225$
Sign $-1$
Analytic cond. $1183.58$
Root an. cond. $34.4032$
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  − 2·2-s − 3·3-s + 2·4-s + 6·6-s + 6·9-s − 6·12-s + 3·13-s − 4·16-s − 3·17-s − 12·18-s − 6·19-s + 4·23-s − 6·26-s − 9·27-s + 29-s + 6·31-s + 8·32-s + 6·34-s + 12·36-s + 12·38-s − 9·39-s − 6·41-s − 6·43-s − 8·46-s + 9·47-s + 12·48-s + 9·51-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 4-s + 2.44·6-s + 2·9-s − 1.73·12-s + 0.832·13-s − 16-s − 0.727·17-s − 2.82·18-s − 1.37·19-s + 0.834·23-s − 1.17·26-s − 1.73·27-s + 0.185·29-s + 1.07·31-s + 1.41·32-s + 1.02·34-s + 2·36-s + 1.94·38-s − 1.44·39-s − 0.937·41-s − 0.914·43-s − 1.17·46-s + 1.31·47-s + 1.73·48-s + 1.26·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(148225\)    =    \(5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1183.58\)
Root analytic conductor: \(34.4032\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 148225,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−13.44767329305248, −13.00877580665404, −12.48404188811187, −11.91569313484851, −11.39450843152607, −11.20372957879197, −10.59265641119437, −10.28447293803948, −10.07348717331703, −9.152232086949019, −8.808236351185664, −8.441454541273588, −7.718634689875633, −7.220674035988054, −6.582634540615074, −6.421531335455676, −5.955023514848441, −4.961620481938696, −4.884971323137708, −4.146737206714879, −3.551788171131878, −2.414392769551230, −1.930013616799341, −1.095504019577497, −0.7156473356207522, 0, 0.7156473356207522, 1.095504019577497, 1.930013616799341, 2.414392769551230, 3.551788171131878, 4.146737206714879, 4.884971323137708, 4.961620481938696, 5.955023514848441, 6.421531335455676, 6.582634540615074, 7.220674035988054, 7.718634689875633, 8.441454541273588, 8.808236351185664, 9.152232086949019, 10.07348717331703, 10.28447293803948, 10.59265641119437, 11.20372957879197, 11.39450843152607, 11.91569313484851, 12.48404188811187, 13.00877580665404, 13.44767329305248

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