Properties

Label 2-93600-1.1-c1-0-80
Degree $2$
Conductor $93600$
Sign $-1$
Analytic cond. $747.399$
Root an. cond. $27.3386$
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  − 3·7-s − 3·11-s + 13-s + 7·17-s + 2·19-s + 23-s − 6·29-s + 10·31-s + 37-s + 41-s − 4·43-s − 4·47-s + 2·49-s − 53-s + 13·61-s − 8·67-s + 5·71-s + 10·73-s + 9·77-s + 79-s − 6·83-s − 9·89-s − 3·91-s − 97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.13·7-s − 0.904·11-s + 0.277·13-s + 1.69·17-s + 0.458·19-s + 0.208·23-s − 1.11·29-s + 1.79·31-s + 0.164·37-s + 0.156·41-s − 0.609·43-s − 0.583·47-s + 2/7·49-s − 0.137·53-s + 1.66·61-s − 0.977·67-s + 0.593·71-s + 1.17·73-s + 1.02·77-s + 0.112·79-s − 0.658·83-s − 0.953·89-s − 0.314·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93600\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(747.399\)
Root analytic conductor: \(27.3386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{93600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 93600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.07183586060835, −13.43467903851560, −13.10017719570418, −12.71067941542514, −12.09748041779314, −11.74127738516546, −11.09265436117007, −10.51654340336640, −9.985688295160043, −9.720628730349367, −9.299201332894781, −8.407769725483267, −8.088092833121331, −7.560939187921375, −6.965215741730072, −6.429468334904792, −5.893570386630613, −5.329139514445612, −4.968986709102954, −3.992543229638699, −3.552201543624424, −2.921308256831546, −2.586083739428018, −1.523515274673647, −0.8429824176093798, 0, 0.8429824176093798, 1.523515274673647, 2.586083739428018, 2.921308256831546, 3.552201543624424, 3.992543229638699, 4.968986709102954, 5.329139514445612, 5.893570386630613, 6.429468334904792, 6.965215741730072, 7.560939187921375, 8.088092833121331, 8.407769725483267, 9.299201332894781, 9.720628730349367, 9.985688295160043, 10.51654340336640, 11.09265436117007, 11.74127738516546, 12.09748041779314, 12.71067941542514, 13.10017719570418, 13.43467903851560, 14.07183586060835

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