Properties

Label 2-360e2-1.1-c1-0-125
Degree $2$
Conductor $129600$
Sign $-1$
Analytic cond. $1034.86$
Root an. cond. $32.1692$
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  − 3·11-s − 4·13-s + 3·17-s − 4·19-s − 2·23-s − 2·29-s − 4·31-s + 4·37-s + 10·41-s + 7·43-s + 4·47-s − 7·49-s + 8·53-s + 59-s − 4·61-s − 12·67-s + 14·71-s + 2·73-s − 83-s − 89-s − 17·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.904·11-s − 1.10·13-s + 0.727·17-s − 0.917·19-s − 0.417·23-s − 0.371·29-s − 0.718·31-s + 0.657·37-s + 1.56·41-s + 1.06·43-s + 0.583·47-s − 49-s + 1.09·53-s + 0.130·59-s − 0.512·61-s − 1.46·67-s + 1.66·71-s + 0.234·73-s − 0.109·83-s − 0.105·89-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129600\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(1034.86\)
Root analytic conductor: \(32.1692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 129600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
good7 \( 1 + p T^{2} \) 1.7.a
11 \( 1 + 3 T + p T^{2} \) 1.11.d
13 \( 1 + 4 T + p T^{2} \) 1.13.e
17 \( 1 - 3 T + p T^{2} \) 1.17.ad
19 \( 1 + 4 T + p T^{2} \) 1.19.e
23 \( 1 + 2 T + p T^{2} \) 1.23.c
29 \( 1 + 2 T + p T^{2} \) 1.29.c
31 \( 1 + 4 T + p T^{2} \) 1.31.e
37 \( 1 - 4 T + p T^{2} \) 1.37.ae
41 \( 1 - 10 T + p T^{2} \) 1.41.ak
43 \( 1 - 7 T + p T^{2} \) 1.43.ah
47 \( 1 - 4 T + p T^{2} \) 1.47.ae
53 \( 1 - 8 T + p T^{2} \) 1.53.ai
59 \( 1 - T + p T^{2} \) 1.59.ab
61 \( 1 + 4 T + p T^{2} \) 1.61.e
67 \( 1 + 12 T + p T^{2} \) 1.67.m
71 \( 1 - 14 T + p T^{2} \) 1.71.ao
73 \( 1 - 2 T + p T^{2} \) 1.73.ac
79 \( 1 + p T^{2} \) 1.79.a
83 \( 1 + T + p T^{2} \) 1.83.b
89 \( 1 + T + p T^{2} \) 1.89.b
97 \( 1 + 17 T + p T^{2} \) 1.97.r
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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

−13.75957664901538, −13.15603144700429, −12.65229960812869, −12.43557403101920, −11.93167002395053, −11.17656173413829, −10.86374313728029, −10.37038480165502, −9.839019774423798, −9.412458914434987, −8.945320865919796, −8.194139653309643, −7.803202982675730, −7.433522484089193, −6.894812268579483, −6.178198966357788, −5.639314173844314, −5.319507630056483, −4.539176335915652, −4.178745350844117, −3.482440497870224, −2.630697272846672, −2.444947077807837, −1.656575197466475, −0.7258872010834401, 0, 0.7258872010834401, 1.656575197466475, 2.444947077807837, 2.630697272846672, 3.482440497870224, 4.178745350844117, 4.539176335915652, 5.319507630056483, 5.639314173844314, 6.178198966357788, 6.894812268579483, 7.433522484089193, 7.803202982675730, 8.194139653309643, 8.945320865919796, 9.412458914434987, 9.839019774423798, 10.37038480165502, 10.86374313728029, 11.17656173413829, 11.93167002395053, 12.43557403101920, 12.65229960812869, 13.15603144700429, 13.75957664901538

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