Properties

Label 2-252e2-1.1-c1-0-50
Degree $2$
Conductor $63504$
Sign $-1$
Analytic cond. $507.081$
Root an. cond. $22.5184$
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  + 4·13-s − 6·17-s + 2·19-s + 3·23-s − 5·25-s − 6·29-s + 5·31-s + 8·37-s − 3·41-s − 2·43-s − 3·47-s + 6·53-s + 12·59-s − 8·61-s − 8·67-s + 15·71-s − 11·73-s + 79-s − 9·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.10·13-s − 1.45·17-s + 0.458·19-s + 0.625·23-s − 25-s − 1.11·29-s + 0.898·31-s + 1.31·37-s − 0.468·41-s − 0.304·43-s − 0.437·47-s + 0.824·53-s + 1.56·59-s − 1.02·61-s − 0.977·67-s + 1.78·71-s − 1.28·73-s + 0.112·79-s − 0.953·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63504\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(507.081\)
Root analytic conductor: \(22.5184\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 63504,\ (\ :1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + p T^{2} \) 1.5.a
11 \( 1 + p T^{2} \) 1.11.a
13 \( 1 - 4 T + p T^{2} \) 1.13.ae
17 \( 1 + 6 T + p T^{2} \) 1.17.g
19 \( 1 - 2 T + p T^{2} \) 1.19.ac
23 \( 1 - 3 T + p T^{2} \) 1.23.ad
29 \( 1 + 6 T + p T^{2} \) 1.29.g
31 \( 1 - 5 T + p T^{2} \) 1.31.af
37 \( 1 - 8 T + p T^{2} \) 1.37.ai
41 \( 1 + 3 T + p T^{2} \) 1.41.d
43 \( 1 + 2 T + p T^{2} \) 1.43.c
47 \( 1 + 3 T + p T^{2} \) 1.47.d
53 \( 1 - 6 T + p T^{2} \) 1.53.ag
59 \( 1 - 12 T + p T^{2} \) 1.59.am
61 \( 1 + 8 T + p T^{2} \) 1.61.i
67 \( 1 + 8 T + p T^{2} \) 1.67.i
71 \( 1 - 15 T + p T^{2} \) 1.71.ap
73 \( 1 + 11 T + p T^{2} \) 1.73.l
79 \( 1 - T + p T^{2} \) 1.79.ab
83 \( 1 + p T^{2} \) 1.83.a
89 \( 1 + 9 T + p T^{2} \) 1.89.j
97 \( 1 + 2 T + p T^{2} \) 1.97.c
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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

−14.55460217022396, −13.77519591027930, −13.48019106328165, −13.14824403284864, −12.62074128812265, −11.66758820064424, −11.61032729317058, −10.98208746149514, −10.56857296603823, −9.816587591679247, −9.405548180153860, −8.850559258173237, −8.330475379965028, −7.880896920726099, −7.123017766157294, −6.707153692525089, −6.064989227429993, −5.639060513632470, −4.915804445703912, −4.240346550223138, −3.836448892975387, −3.081539258029695, −2.408715381455671, −1.702778380725421, −0.9576699976876822, 0, 0.9576699976876822, 1.702778380725421, 2.408715381455671, 3.081539258029695, 3.836448892975387, 4.240346550223138, 4.915804445703912, 5.639060513632470, 6.064989227429993, 6.707153692525089, 7.123017766157294, 7.880896920726099, 8.330475379965028, 8.850559258173237, 9.405548180153860, 9.816587591679247, 10.56857296603823, 10.98208746149514, 11.61032729317058, 11.66758820064424, 12.62074128812265, 13.14824403284864, 13.48019106328165, 13.77519591027930, 14.55460217022396

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