Properties

Label 2-85176-1.1-c1-0-50
Degree $2$
Conductor $85176$
Sign $-1$
Analytic cond. $680.133$
Root an. cond. $26.0793$
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 + 2·11-s + 2·17-s − 5·25-s + 6·29-s − 4·31-s + 2·37-s + 12·43-s + 2·47-s + 49-s − 6·53-s − 6·59-s − 14·61-s + 12·67-s − 10·71-s + 10·73-s + 2·77-s − 2·83-s − 4·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 2·119-s + ⋯
L(s)  = 1  + 0.377·7-s + 0.603·11-s + 0.485·17-s − 25-s + 1.11·29-s − 0.718·31-s + 0.328·37-s + 1.82·43-s + 0.291·47-s + 1/7·49-s − 0.824·53-s − 0.781·59-s − 1.79·61-s + 1.46·67-s − 1.18·71-s + 1.17·73-s + 0.227·77-s − 0.219·83-s − 0.423·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.183·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85176\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(680.133\)
Root analytic conductor: \(26.0793\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 85176,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
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 \( 1 + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.21293859440197, −13.80933443529452, −13.22227215426477, −12.56582161213961, −12.18821191076570, −11.80749833167526, −11.09400468153357, −10.83934501294155, −10.18611068222417, −9.616404150547915, −9.188361114581150, −8.718152534993989, −7.965364445955816, −7.686402030784983, −7.137320404721428, −6.297619533274103, −6.115020020311570, −5.341685646009655, −4.834784321149529, −4.121103951822117, −3.773675783257445, −2.939257772820024, −2.372126794994414, −1.547498965782213, −1.025532912705422, 0, 1.025532912705422, 1.547498965782213, 2.372126794994414, 2.939257772820024, 3.773675783257445, 4.121103951822117, 4.834784321149529, 5.341685646009655, 6.115020020311570, 6.297619533274103, 7.137320404721428, 7.686402030784983, 7.965364445955816, 8.718152534993989, 9.188361114581150, 9.616404150547915, 10.18611068222417, 10.83934501294155, 11.09400468153357, 11.80749833167526, 12.18821191076570, 12.56582161213961, 13.22227215426477, 13.80933443529452, 14.21293859440197

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