Properties

Label 2-234416-1.1-c1-0-24
Degree $2$
Conductor $234416$
Sign $-1$
Analytic cond. $1871.82$
Root an. cond. $43.2645$
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  − 3·9-s + 13-s − 6·17-s − 4·19-s + 23-s − 5·25-s − 2·29-s − 10·31-s + 4·37-s − 10·41-s + 4·43-s + 6·47-s − 2·53-s + 2·59-s − 10·61-s + 16·67-s − 10·71-s + 2·73-s + 8·79-s + 9·81-s + 16·83-s + 8·89-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 9-s + 0.277·13-s − 1.45·17-s − 0.917·19-s + 0.208·23-s − 25-s − 0.371·29-s − 1.79·31-s + 0.657·37-s − 1.56·41-s + 0.609·43-s + 0.875·47-s − 0.274·53-s + 0.260·59-s − 1.28·61-s + 1.95·67-s − 1.18·71-s + 0.234·73-s + 0.900·79-s + 81-s + 1.75·83-s + 0.847·89-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234416\)    =    \(2^{4} \cdot 7^{2} \cdot 13 \cdot 23\)
Sign: $-1$
Analytic conductor: \(1871.82\)
Root analytic conductor: \(43.2645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{234416} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 234416,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
23 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−13.21863720107121, −12.75583330451870, −12.12656237383949, −11.78218949069534, −11.09349817278852, −10.96290099486733, −10.57494251645890, −9.828618234774585, −9.181098599481609, −9.045578861199797, −8.516695364010681, −7.979539733974537, −7.569930311736677, −6.829437944590068, −6.532053826626046, −5.870436114499893, −5.613762569595675, −4.877778739569011, −4.447382555938977, −3.663055136967486, −3.516871501944114, −2.536402518939033, −2.166787991659132, −1.668015581952756, −0.5785682734803539, 0, 0.5785682734803539, 1.668015581952756, 2.166787991659132, 2.536402518939033, 3.516871501944114, 3.663055136967486, 4.447382555938977, 4.877778739569011, 5.613762569595675, 5.870436114499893, 6.532053826626046, 6.829437944590068, 7.569930311736677, 7.979539733974537, 8.516695364010681, 9.045578861199797, 9.181098599481609, 9.828618234774585, 10.57494251645890, 10.96290099486733, 11.09349817278852, 11.78218949069534, 12.12656237383949, 12.75583330451870, 13.21863720107121

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