Properties

Label 2-1840-1.1-c3-0-0
Degree $2$
Conductor $1840$
Sign $1$
Analytic cond. $108.563$
Root an. cond. $10.4193$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.52·3-s − 5·5-s − 24.3·7-s − 20.6·9-s − 65.7·11-s + 11.1·13-s − 12.6·15-s − 128.·17-s − 131.·19-s − 61.5·21-s − 23·23-s + 25·25-s − 120.·27-s − 50.1·29-s + 96.2·31-s − 166.·33-s + 121.·35-s + 74.8·37-s + 28.1·39-s − 6.78·41-s + 11.7·43-s + 103.·45-s + 236.·47-s + 249.·49-s − 325.·51-s − 123.·53-s + 328.·55-s + ⋯
L(s)  = 1  + 0.486·3-s − 0.447·5-s − 1.31·7-s − 0.762·9-s − 1.80·11-s + 0.237·13-s − 0.217·15-s − 1.83·17-s − 1.58·19-s − 0.639·21-s − 0.208·23-s + 0.200·25-s − 0.858·27-s − 0.321·29-s + 0.557·31-s − 0.877·33-s + 0.587·35-s + 0.332·37-s + 0.115·39-s − 0.0258·41-s + 0.0417·43-s + 0.341·45-s + 0.734·47-s + 0.727·49-s − 0.893·51-s − 0.321·53-s + 0.806·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(108.563\)
Root analytic conductor: \(10.4193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.06993605889\)
\(L(\frac12)\) \(\approx\) \(0.06993605889\)
\(L(\frac{5}{2})\) 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 \( 1 + 5T \)
23 \( 1 + 23T \)
good3 \( 1 - 2.52T + 27T^{2} \)
7 \( 1 + 24.3T + 343T^{2} \)
11 \( 1 + 65.7T + 1.33e3T^{2} \)
13 \( 1 - 11.1T + 2.19e3T^{2} \)
17 \( 1 + 128.T + 4.91e3T^{2} \)
19 \( 1 + 131.T + 6.85e3T^{2} \)
29 \( 1 + 50.1T + 2.43e4T^{2} \)
31 \( 1 - 96.2T + 2.97e4T^{2} \)
37 \( 1 - 74.8T + 5.06e4T^{2} \)
41 \( 1 + 6.78T + 6.89e4T^{2} \)
43 \( 1 - 11.7T + 7.95e4T^{2} \)
47 \( 1 - 236.T + 1.03e5T^{2} \)
53 \( 1 + 123.T + 1.48e5T^{2} \)
59 \( 1 - 499.T + 2.05e5T^{2} \)
61 \( 1 - 167.T + 2.26e5T^{2} \)
67 \( 1 + 1.06e3T + 3.00e5T^{2} \)
71 \( 1 + 956.T + 3.57e5T^{2} \)
73 \( 1 - 353.T + 3.89e5T^{2} \)
79 \( 1 + 420.T + 4.93e5T^{2} \)
83 \( 1 - 1.03e3T + 5.71e5T^{2} \)
89 \( 1 - 657.T + 7.04e5T^{2} \)
97 \( 1 + 1.37e3T + 9.12e5T^{2} \)
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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

−8.726406485557957872883479755624, −8.300431718970757320048035754008, −7.37690473683126021289280865924, −6.47501887414163109305036283732, −5.83633204371367376577606979635, −4.68485815028595867060551781678, −3.80024560592852300287734488096, −2.75519854190693122683526144579, −2.33256394110839768741603603326, −0.10876813697882720780066962269, 0.10876813697882720780066962269, 2.33256394110839768741603603326, 2.75519854190693122683526144579, 3.80024560592852300287734488096, 4.68485815028595867060551781678, 5.83633204371367376577606979635, 6.47501887414163109305036283732, 7.37690473683126021289280865924, 8.300431718970757320048035754008, 8.726406485557957872883479755624

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