Properties

Label 2-355740-1.1-c1-0-79
Degree $2$
Conductor $355740$
Sign $-1$
Analytic cond. $2840.59$
Root an. cond. $53.2972$
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-s + 5-s + 9-s + 2·13-s − 15-s − 6·17-s + 8·19-s − 6·23-s + 25-s − 27-s − 6·29-s − 2·31-s + 2·37-s − 2·39-s − 8·43-s + 45-s + 12·47-s + 6·51-s + 6·53-s − 8·57-s − 6·59-s + 8·61-s + 2·65-s + 2·67-s + 6·69-s − 10·73-s − 75-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 1.45·17-s + 1.83·19-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.328·37-s − 0.320·39-s − 1.21·43-s + 0.149·45-s + 1.75·47-s + 0.840·51-s + 0.824·53-s − 1.05·57-s − 0.781·59-s + 1.02·61-s + 0.248·65-s + 0.244·67-s + 0.722·69-s − 1.17·73-s − 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 355740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(355740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(2840.59\)
Root analytic conductor: \(53.2972\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{355740} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 355740,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.79231484641798, −12.20506157164795, −11.81842890833216, −11.31058041207496, −11.15726470777642, −10.37959730401098, −10.19879224248476, −9.532079892323534, −9.233491118829809, −8.731549572040837, −8.209440028905548, −7.596162614627563, −7.151167193617870, −6.809830087726165, −6.088588786474378, −5.775858106559353, −5.449228883141049, −4.756959440615196, −4.329682395013205, −3.697082162550334, −3.305720827018113, −2.443134965378003, −2.017180274264777, −1.404922510973530, −0.7410029173373133, 0, 0.7410029173373133, 1.404922510973530, 2.017180274264777, 2.443134965378003, 3.305720827018113, 3.697082162550334, 4.329682395013205, 4.756959440615196, 5.449228883141049, 5.775858106559353, 6.088588786474378, 6.809830087726165, 7.151167193617870, 7.596162614627563, 8.209440028905548, 8.731549572040837, 9.233491118829809, 9.532079892323534, 10.19879224248476, 10.37959730401098, 11.15726470777642, 11.31058041207496, 11.81842890833216, 12.20506157164795, 12.79231484641798

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