Properties

Label 2-85176-1.1-c1-0-40
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  − 2·5-s + 7-s + 3·11-s + 17-s − 5·19-s − 2·23-s − 25-s − 29-s + 2·31-s − 2·35-s − 2·37-s + 5·41-s + 2·43-s − 3·47-s + 49-s + 9·53-s − 6·55-s − 6·59-s + 13·61-s − 2·67-s − 12·71-s + 8·73-s + 3·77-s − 17·79-s − 2·85-s − 5·89-s + 10·95-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s + 0.904·11-s + 0.242·17-s − 1.14·19-s − 0.417·23-s − 1/5·25-s − 0.185·29-s + 0.359·31-s − 0.338·35-s − 0.328·37-s + 0.780·41-s + 0.304·43-s − 0.437·47-s + 1/7·49-s + 1.23·53-s − 0.809·55-s − 0.781·59-s + 1.66·61-s − 0.244·67-s − 1.42·71-s + 0.936·73-s + 0.341·77-s − 1.91·79-s − 0.216·85-s − 0.529·89-s + 1.02·95-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: $\chi_{85176} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
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

−14.27442340250010, −13.76292534202734, −13.09053582221821, −12.67787333496838, −11.98982944621207, −11.82671477638410, −11.26037318442240, −10.80986170410600, −10.20749757102643, −9.721938430358512, −9.020296909688335, −8.617362514787215, −8.112377996916491, −7.661022181496977, −7.023960799382872, −6.623494258542199, −5.886716628501138, −5.488616037860862, −4.520515413911813, −4.244595415478906, −3.781335217156548, −3.081704929681520, −2.289414796360923, −1.647254506349772, −0.8463342136399954, 0, 0.8463342136399954, 1.647254506349772, 2.289414796360923, 3.081704929681520, 3.781335217156548, 4.244595415478906, 4.520515413911813, 5.488616037860862, 5.886716628501138, 6.623494258542199, 7.023960799382872, 7.661022181496977, 8.112377996916491, 8.617362514787215, 9.020296909688335, 9.721938430358512, 10.20749757102643, 10.80986170410600, 11.26037318442240, 11.82671477638410, 11.98982944621207, 12.67787333496838, 13.09053582221821, 13.76292534202734, 14.27442340250010

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