Properties

Label 2-85176-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 4·11-s − 6·17-s + 4·19-s + 4·23-s − 5·25-s + 2·29-s − 8·31-s − 6·37-s + 6·41-s + 4·43-s + 6·47-s + 49-s + 6·53-s − 12·59-s + 8·61-s − 6·67-s − 12·71-s + 2·73-s + 4·77-s − 12·79-s − 10·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 1.45·17-s + 0.917·19-s + 0.834·23-s − 25-s + 0.371·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s + 1.02·61-s − 0.733·67-s − 1.42·71-s + 0.234·73-s + 0.455·77-s − 1.35·79-s − 1.05·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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: \(0\)
Selberg data: \((2,\ 85176,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7499523606\)
\(L(\frac12)\) \(\approx\) \(0.7499523606\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.75913370707442, −13.41828290690935, −13.02398355997854, −12.51521051417218, −12.00457746768632, −11.35440058652617, −10.91233567235375, −10.53664177475750, −9.952495623323725, −9.377150589547651, −8.903596566860741, −8.516758216972748, −7.679387574286379, −7.308914645542721, −6.969402740051025, −6.070987944343457, −5.700532897504553, −5.148085279920903, −4.520687419907008, −3.948262398405824, −3.235654538250052, −2.648736592375707, −2.138509841515715, −1.286429373385798, −0.2788778448218502, 0.2788778448218502, 1.286429373385798, 2.138509841515715, 2.648736592375707, 3.235654538250052, 3.948262398405824, 4.520687419907008, 5.148085279920903, 5.700532897504553, 6.070987944343457, 6.969402740051025, 7.308914645542721, 7.679387574286379, 8.516758216972748, 8.903596566860741, 9.377150589547651, 9.952495623323725, 10.53664177475750, 10.91233567235375, 11.35440058652617, 12.00457746768632, 12.51521051417218, 13.02398355997854, 13.41828290690935, 13.75913370707442

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