Properties

Label 2-301180-1.1-c1-0-9
Degree $2$
Conductor $301180$
Sign $1$
Analytic cond. $2404.93$
Root an. cond. $49.0401$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s − 4·7-s + 9-s − 11-s + 4·13-s + 2·15-s + 4·19-s + 8·21-s + 6·23-s + 25-s + 4·27-s + 6·29-s − 8·31-s + 2·33-s + 4·35-s − 8·39-s + 6·41-s − 8·43-s − 45-s + 6·47-s + 9·49-s − 6·53-s + 55-s − 8·57-s + 12·59-s − 2·61-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.516·15-s + 0.917·19-s + 1.74·21-s + 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 1.43·31-s + 0.348·33-s + 0.676·35-s − 1.28·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s + 0.875·47-s + 9/7·49-s − 0.824·53-s + 0.134·55-s − 1.05·57-s + 1.56·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 301180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(301180\)    =    \(2^{2} \cdot 5 \cdot 11 \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(2404.93\)
Root analytic conductor: \(49.0401\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{301180} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 301180,\ (\ :1/2),\ 1)\)

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 \( 1 \)
5 \( 1 + T \)
11 \( 1 + T \)
37 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.07893076051328, −12.59882647077194, −12.22587079831857, −11.77688083643415, −11.21067685737856, −11.02421575158807, −10.41203082734855, −10.11597221786042, −9.522196951694045, −8.911655682118551, −8.747471479011251, −8.016033442297515, −7.327607793711959, −6.996586770720735, −6.554478989657221, −6.096303481397097, −5.528769632739463, −5.378383636012697, −4.570158585389574, −4.058495662662019, −3.477667194863838, −2.907605075339168, −2.707813369702774, −1.282875539885425, −1.138904137249732, 0, 0, 1.138904137249732, 1.282875539885425, 2.707813369702774, 2.907605075339168, 3.477667194863838, 4.058495662662019, 4.570158585389574, 5.378383636012697, 5.528769632739463, 6.096303481397097, 6.554478989657221, 6.996586770720735, 7.327607793711959, 8.016033442297515, 8.747471479011251, 8.911655682118551, 9.522196951694045, 10.11597221786042, 10.41203082734855, 11.02421575158807, 11.21067685737856, 11.77688083643415, 12.22587079831857, 12.59882647077194, 13.07893076051328

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