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Properties

Label 2-46800-1.1-c1-0-120
Degree $2$
Conductor $46800$
Sign $-1$
Analytic cond. $373.699$
Root an. cond. $19.3313$
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  − 13-s + 2·17-s + 8·19-s + 4·23-s + 2·29-s + 4·31-s − 6·37-s − 10·41-s − 8·47-s − 7·49-s + 6·53-s + 8·59-s − 2·61-s + 4·67-s − 12·71-s − 10·73-s + 8·79-s − 12·83-s − 10·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.277·13-s + 0.485·17-s + 1.83·19-s + 0.834·23-s + 0.371·29-s + 0.718·31-s − 0.986·37-s − 1.56·41-s − 1.16·47-s − 49-s + 0.824·53-s + 1.04·59-s − 0.256·61-s + 0.488·67-s − 1.42·71-s − 1.17·73-s + 0.900·79-s − 1.31·83-s − 1.05·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46800\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(373.699\)
Root analytic conductor: \(19.3313\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ -1)\)

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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.85437279575410, −14.22604406665784, −13.97002260579796, −13.14885534783424, −13.04892258070270, −12.06516671514127, −11.77458770555059, −11.48657756224996, −10.53405563017253, −10.21854901563465, −9.635969567831566, −9.204813698265641, −8.417977381645676, −8.106788877158086, −7.274229989107418, −6.997728329322204, −6.342816890391104, −5.496230307868748, −5.170105641366627, −4.613685793434742, −3.701540826220704, −3.159181939835452, −2.681441723772868, −1.594785246701786, −1.088364592261929, 0, 1.088364592261929, 1.594785246701786, 2.681441723772868, 3.159181939835452, 3.701540826220704, 4.613685793434742, 5.170105641366627, 5.496230307868748, 6.342816890391104, 6.997728329322204, 7.274229989107418, 8.106788877158086, 8.417977381645676, 9.204813698265641, 9.635969567831566, 10.21854901563465, 10.53405563017253, 11.48657756224996, 11.77458770555059, 12.06516671514127, 13.04892258070270, 13.14885534783424, 13.97002260579796, 14.22604406665784, 14.85437279575410

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