Properties

Label 2-1840-1.1-c1-0-6
Degree $2$
Conductor $1840$
Sign $1$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.568·3-s − 5-s − 4.73·7-s − 2.67·9-s + 0.360·11-s + 5.26·13-s − 0.568·15-s + 0.370·17-s + 4.60·19-s − 2.69·21-s + 23-s + 25-s − 3.22·27-s + 0.939·29-s − 9.66·31-s + 0.204·33-s + 4.73·35-s + 3.26·37-s + 2.99·39-s + 5.29·41-s + 2.67·45-s + 1.25·47-s + 15.4·49-s + 0.210·51-s + 10.9·53-s − 0.360·55-s + 2.61·57-s + ⋯
L(s)  = 1  + 0.328·3-s − 0.447·5-s − 1.79·7-s − 0.892·9-s + 0.108·11-s + 1.45·13-s − 0.146·15-s + 0.0899·17-s + 1.05·19-s − 0.587·21-s + 0.208·23-s + 0.200·25-s − 0.620·27-s + 0.174·29-s − 1.73·31-s + 0.0356·33-s + 0.800·35-s + 0.537·37-s + 0.478·39-s + 0.827·41-s + 0.399·45-s + 0.183·47-s + 2.20·49-s + 0.0295·51-s + 1.50·53-s − 0.0486·55-s + 0.346·57-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.281066291\)
\(L(\frac12)\) \(\approx\) \(1.281066291\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 - 0.568T + 3T^{2} \)
7 \( 1 + 4.73T + 7T^{2} \)
11 \( 1 - 0.360T + 11T^{2} \)
13 \( 1 - 5.26T + 13T^{2} \)
17 \( 1 - 0.370T + 17T^{2} \)
19 \( 1 - 4.60T + 19T^{2} \)
29 \( 1 - 0.939T + 29T^{2} \)
31 \( 1 + 9.66T + 31T^{2} \)
37 \( 1 - 3.26T + 37T^{2} \)
41 \( 1 - 5.29T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 1.25T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 - 9.66T + 59T^{2} \)
61 \( 1 - 9.71T + 61T^{2} \)
67 \( 1 - 7.07T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 0.745T + 73T^{2} \)
79 \( 1 + 0.415T + 79T^{2} \)
83 \( 1 - 9.26T + 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−9.109959738712378489852191146317, −8.693304939406647064969092670324, −7.66031533688569428577097904332, −6.86974555955036521775890010158, −6.03835507846753314531492757952, −5.45116032660235916260362165796, −3.78412579682264420598315908340, −3.50498762785343705309704084704, −2.54526778852463296582397838950, −0.73930114888484845428126629533, 0.73930114888484845428126629533, 2.54526778852463296582397838950, 3.50498762785343705309704084704, 3.78412579682264420598315908340, 5.45116032660235916260362165796, 6.03835507846753314531492757952, 6.86974555955036521775890010158, 7.66031533688569428577097904332, 8.693304939406647064969092670324, 9.109959738712378489852191146317

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