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Properties

Label 2-226512-1.1-c1-0-134
Degree $2$
Conductor $226512$
Sign $-1$
Analytic cond. $1808.70$
Root an. cond. $42.5289$
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  + 4·5-s − 7-s − 13-s − 8·17-s − 5·19-s + 4·23-s + 11·25-s − 4·29-s + 9·31-s − 4·35-s + 7·37-s + 8·41-s − 8·43-s + 8·47-s − 6·49-s − 8·53-s + 8·59-s + 7·61-s − 4·65-s + 67-s − 12·71-s + 5·73-s − 11·79-s − 12·83-s − 32·85-s − 8·89-s + 91-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.377·7-s − 0.277·13-s − 1.94·17-s − 1.14·19-s + 0.834·23-s + 11/5·25-s − 0.742·29-s + 1.61·31-s − 0.676·35-s + 1.15·37-s + 1.24·41-s − 1.21·43-s + 1.16·47-s − 6/7·49-s − 1.09·53-s + 1.04·59-s + 0.896·61-s − 0.496·65-s + 0.122·67-s − 1.42·71-s + 0.585·73-s − 1.23·79-s − 1.31·83-s − 3.47·85-s − 0.847·89-s + 0.104·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(226512\)    =    \(2^{4} \cdot 3^{2} \cdot 11^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1808.70\)
Root analytic conductor: \(42.5289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 226512,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 \)
13 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−13.22902309742738, −12.93685816586909, −12.51800800291105, −11.66725032141753, −11.21716854067088, −10.82573515646371, −10.30955305785634, −9.876760194297676, −9.443313816021076, −9.093362888097233, −8.547899098554862, −8.225548569014377, −7.250363681221908, −6.834007306913862, −6.476276716999204, −5.974776767053898, −5.689321149212157, −4.785089996893599, −4.617971216725378, −3.996566490807287, −3.026444013604093, −2.558222533106761, −2.228675869903742, −1.617568777350902, −0.8820968748213705, 0, 0.8820968748213705, 1.617568777350902, 2.228675869903742, 2.558222533106761, 3.026444013604093, 3.996566490807287, 4.617971216725378, 4.785089996893599, 5.689321149212157, 5.974776767053898, 6.476276716999204, 6.834007306913862, 7.250363681221908, 8.225548569014377, 8.547899098554862, 9.093362888097233, 9.443313816021076, 9.876760194297676, 10.30955305785634, 10.82573515646371, 11.21716854067088, 11.66725032141753, 12.51800800291105, 12.93685816586909, 13.22902309742738

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