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Properties

Label 2-80275-1.1-c1-0-20
Degree $2$
Conductor $80275$
Sign $-1$
Analytic cond. $640.999$
Root an. cond. $25.3179$
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  + 2·3-s − 2·4-s − 7-s + 9-s − 3·11-s − 4·12-s + 4·16-s + 3·17-s − 19-s − 2·21-s − 4·27-s + 2·28-s + 6·29-s + 4·31-s − 6·33-s − 2·36-s + 2·37-s + 6·41-s + 43-s + 6·44-s − 3·47-s + 8·48-s − 6·49-s + 6·51-s − 12·53-s − 2·57-s + 6·59-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.15·12-s + 16-s + 0.727·17-s − 0.229·19-s − 0.436·21-s − 0.769·27-s + 0.377·28-s + 1.11·29-s + 0.718·31-s − 1.04·33-s − 1/3·36-s + 0.328·37-s + 0.937·41-s + 0.152·43-s + 0.904·44-s − 0.437·47-s + 1.15·48-s − 6/7·49-s + 0.840·51-s − 1.64·53-s − 0.264·57-s + 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80275\)    =    \(5^{2} \cdot 13^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(640.999\)
Root analytic conductor: \(25.3179\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 80275,\ (\ :1/2),\ -1)\)

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

−14.16454956380461, −13.84788173419982, −13.28845124603192, −12.86690063348558, −12.55641223355882, −11.86691201555840, −11.23832637508429, −10.50378046259531, −10.04757506001136, −9.708772764351357, −9.140646914455641, −8.678967105489761, −8.200073406416921, −7.772444165821690, −7.445208179297350, −6.391566019103084, −6.045904845069805, −5.235064022432633, −4.801432246074309, −4.172799877898715, −3.543992572874344, −2.963691488212749, −2.660793993858019, −1.715251841305294, −0.8610105417764594, 0, 0.8610105417764594, 1.715251841305294, 2.660793993858019, 2.963691488212749, 3.543992572874344, 4.172799877898715, 4.801432246074309, 5.235064022432633, 6.045904845069805, 6.391566019103084, 7.445208179297350, 7.772444165821690, 8.200073406416921, 8.678967105489761, 9.140646914455641, 9.708772764351357, 10.04757506001136, 10.50378046259531, 11.23832637508429, 11.86691201555840, 12.55641223355882, 12.86690063348558, 13.28845124603192, 13.84788173419982, 14.16454956380461

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