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Properties

Label 2-1840-23.22-c2-0-18
Degree $2$
Conductor $1840$
Sign $-0.771 - 0.636i$
Analytic cond. $50.1363$
Root an. cond. $7.08070$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.53·3-s + 2.23i·5-s + 5.73i·7-s + 11.5·9-s − 14.4i·11-s − 22.5·13-s + 10.1i·15-s + 15.8i·17-s + 21.5i·19-s + 25.9i·21-s + (−14.6 + 17.7i)23-s − 5.00·25-s + 11.5·27-s − 23.3·29-s − 52.1·31-s + ⋯
L(s)  = 1  + 1.51·3-s + 0.447i·5-s + 0.818i·7-s + 1.28·9-s − 1.31i·11-s − 1.73·13-s + 0.675i·15-s + 0.935i·17-s + 1.13i·19-s + 1.23i·21-s + (−0.636 + 0.771i)23-s − 0.200·25-s + 0.427·27-s − 0.804·29-s − 1.68·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.771 - 0.636i$
Analytic conductor: \(50.1363\)
Root analytic conductor: \(7.08070\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1),\ -0.771 - 0.636i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.757847161\)
\(L(\frac12)\) \(\approx\) \(1.757847161\)
\(L(2)\) 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 \( 1 - 2.23iT \)
23 \( 1 + (14.6 - 17.7i)T \)
good3 \( 1 - 4.53T + 9T^{2} \)
7 \( 1 - 5.73iT - 49T^{2} \)
11 \( 1 + 14.4iT - 121T^{2} \)
13 \( 1 + 22.5T + 169T^{2} \)
17 \( 1 - 15.8iT - 289T^{2} \)
19 \( 1 - 21.5iT - 361T^{2} \)
29 \( 1 + 23.3T + 841T^{2} \)
31 \( 1 + 52.1T + 961T^{2} \)
37 \( 1 + 29.3iT - 1.36e3T^{2} \)
41 \( 1 - 69.3T + 1.68e3T^{2} \)
43 \( 1 - 59.0iT - 1.84e3T^{2} \)
47 \( 1 + 35.0T + 2.20e3T^{2} \)
53 \( 1 - 55.7iT - 2.80e3T^{2} \)
59 \( 1 + 34.7T + 3.48e3T^{2} \)
61 \( 1 + 35.4iT - 3.72e3T^{2} \)
67 \( 1 - 111. iT - 4.48e3T^{2} \)
71 \( 1 - 94.6T + 5.04e3T^{2} \)
73 \( 1 - 8.86T + 5.32e3T^{2} \)
79 \( 1 + 86.9iT - 6.24e3T^{2} \)
83 \( 1 - 153. iT - 6.88e3T^{2} \)
89 \( 1 + 152. iT - 7.92e3T^{2} \)
97 \( 1 - 14.2iT - 9.40e3T^{2} \)
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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

−9.324408534728575141154730243580, −8.604707946293409605917388029136, −7.76367914923656839708944173878, −7.48165406966651184889153310518, −6.06039998862707545962437225885, −5.52266506582617807513287258981, −4.05016557363557505975902699259, −3.36719896799708741661259231958, −2.52922977437871939268527828412, −1.79764324631340139512184023837, 0.31752311491822613000389545392, 1.99337453050230341752778572340, 2.51962123062046922456987951303, 3.71258742804942264645114068433, 4.54849271518732415727710953363, 5.13877658369106577230226371062, 6.86190792828059244501508425552, 7.40881402781484752766065386934, 7.78386414828256014754002888220, 8.932650328881497780813096335890

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