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Properties

Label 2-2040-17.13-c1-0-10
Degree $2$
Conductor $2040$
Sign $0.448 - 0.893i$
Analytic cond. $16.2894$
Root an. cond. $4.03602$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s + (1.99 − 1.99i)7-s + 1.00i·9-s + (−0.973 + 0.973i)11-s − 5.30·13-s − 1.00i·15-s + (4.04 − 0.811i)17-s + 7.91i·19-s − 2.82·21-s + (−6.39 + 6.39i)23-s + 1.00i·25-s + (0.707 − 0.707i)27-s + (2.61 + 2.61i)29-s + (5.62 + 5.62i)31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.316 + 0.316i)5-s + (0.753 − 0.753i)7-s + 0.333i·9-s + (−0.293 + 0.293i)11-s − 1.47·13-s − 0.258i·15-s + (0.980 − 0.196i)17-s + 1.81i·19-s − 0.615·21-s + (−1.33 + 1.33i)23-s + 0.200i·25-s + (0.136 − 0.136i)27-s + (0.485 + 0.485i)29-s + (1.00 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.448 - 0.893i$
Analytic conductor: \(16.2894\)
Root analytic conductor: \(4.03602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2040} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2040,\ (\ :1/2),\ 0.448 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.274466086\)
\(L(\frac12)\) \(\approx\) \(1.274466086\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (-4.04 + 0.811i)T \)
good7 \( 1 + (-1.99 + 1.99i)T - 7iT^{2} \)
11 \( 1 + (0.973 - 0.973i)T - 11iT^{2} \)
13 \( 1 + 5.30T + 13T^{2} \)
19 \( 1 - 7.91iT - 19T^{2} \)
23 \( 1 + (6.39 - 6.39i)T - 23iT^{2} \)
29 \( 1 + (-2.61 - 2.61i)T + 29iT^{2} \)
31 \( 1 + (-5.62 - 5.62i)T + 31iT^{2} \)
37 \( 1 + (6.31 + 6.31i)T + 37iT^{2} \)
41 \( 1 + (-7.51 + 7.51i)T - 41iT^{2} \)
43 \( 1 - 3.41iT - 43T^{2} \)
47 \( 1 - 0.558T + 47T^{2} \)
53 \( 1 - 9.29iT - 53T^{2} \)
59 \( 1 + 6.38iT - 59T^{2} \)
61 \( 1 + (0.253 - 0.253i)T - 61iT^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + (-3.96 - 3.96i)T + 71iT^{2} \)
73 \( 1 + (-2.81 - 2.81i)T + 73iT^{2} \)
79 \( 1 + (6.24 - 6.24i)T - 79iT^{2} \)
83 \( 1 - 0.373iT - 83T^{2} \)
89 \( 1 + 6.65T + 89T^{2} \)
97 \( 1 + (-13.6 - 13.6i)T + 97iT^{2} \)
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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

−9.485404349318528730360270895862, −8.183501697488648692837182568686, −7.52316433251923374010110416142, −7.23942299205546587084614905610, −5.98255999782095111977550564608, −5.39134043577825757376542510744, −4.50627455303996027895705767338, −3.46886421652948979591771635680, −2.18550225161834474436764065637, −1.28080380164474543274717067327, 0.49566574676040864375387260459, 2.15039799810505167100012461961, 2.89694742099735779511485024049, 4.54287508927529969645900181423, 4.81723019122099698822896845295, 5.72288998701427175627603400830, 6.44982662910162344762826777352, 7.55922710754401360375899384664, 8.310031145593077007295644401332, 8.978668704195481660093517195087

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