Properties

Label 2-2320-1.1-c1-0-51
Degree $2$
Conductor $2320$
Sign $-1$
Analytic cond. $18.5252$
Root an. cond. $4.30410$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.70·3-s + 5-s − 3.70·7-s − 0.0783·9-s + 0.630·11-s − 4.34·13-s + 1.70·15-s − 1.55·17-s + 5.70·19-s − 6.34·21-s − 6.63·23-s + 25-s − 5.26·27-s − 29-s + 2.29·31-s + 1.07·33-s − 3.70·35-s − 2.44·37-s − 7.41·39-s + 5.60·41-s − 12.5·43-s − 0.0783·45-s − 2.29·47-s + 6.75·49-s − 2.65·51-s + 0.921·53-s + 0.630·55-s + ⋯
L(s)  = 1  + 0.986·3-s + 0.447·5-s − 1.40·7-s − 0.0261·9-s + 0.190·11-s − 1.20·13-s + 0.441·15-s − 0.376·17-s + 1.30·19-s − 1.38·21-s − 1.38·23-s + 0.200·25-s − 1.01·27-s − 0.185·29-s + 0.411·31-s + 0.187·33-s − 0.626·35-s − 0.402·37-s − 1.18·39-s + 0.874·41-s − 1.91·43-s − 0.0116·45-s − 0.334·47-s + 0.965·49-s − 0.371·51-s + 0.126·53-s + 0.0850·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2320\)    =    \(2^{4} \cdot 5 \cdot 29\)
Sign: $-1$
Analytic conductor: \(18.5252\)
Root analytic conductor: \(4.30410\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2320,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
29 \( 1 + T \)
good3 \( 1 - 1.70T + 3T^{2} \)
7 \( 1 + 3.70T + 7T^{2} \)
11 \( 1 - 0.630T + 11T^{2} \)
13 \( 1 + 4.34T + 13T^{2} \)
17 \( 1 + 1.55T + 17T^{2} \)
19 \( 1 - 5.70T + 19T^{2} \)
23 \( 1 + 6.63T + 23T^{2} \)
31 \( 1 - 2.29T + 31T^{2} \)
37 \( 1 + 2.44T + 37T^{2} \)
41 \( 1 - 5.60T + 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 + 2.29T + 47T^{2} \)
53 \( 1 - 0.921T + 53T^{2} \)
59 \( 1 - 3.60T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 15.6T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 3.12T + 83T^{2} \)
89 \( 1 - 1.41T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−8.774830824989031870785744852648, −7.83099811292399877432628865849, −7.17607561982663976030219482907, −6.29804611003827923551208339098, −5.58656973697870970192704263805, −4.46502584156946472834945228109, −3.34236184894800141692173368154, −2.87271802482425086336827568496, −1.87917261777697366304654108321, 0, 1.87917261777697366304654108321, 2.87271802482425086336827568496, 3.34236184894800141692173368154, 4.46502584156946472834945228109, 5.58656973697870970192704263805, 6.29804611003827923551208339098, 7.17607561982663976030219482907, 7.83099811292399877432628865849, 8.774830824989031870785744852648

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