Properties

Label 2-1840-460.39-c0-0-1
Degree $2$
Conductor $1840$
Sign $0.130 + 0.991i$
Analytic cond. $0.918279$
Root an. cond. $0.958269$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.27 − 0.817i)3-s + (−0.415 − 0.909i)5-s + (0.540 − 0.158i)7-s + (0.533 − 1.16i)9-s + (−1.27 − 0.817i)15-s + (0.557 − 0.643i)21-s + (−0.755 − 0.654i)23-s + (−0.654 + 0.755i)25-s + (−0.0612 − 0.425i)27-s + (−0.118 + 0.822i)29-s + (−0.368 − 0.425i)35-s + (0.797 + 1.74i)41-s + (1.66 − 1.07i)43-s − 1.28·45-s − 1.81·47-s + ⋯
L(s)  = 1  + (1.27 − 0.817i)3-s + (−0.415 − 0.909i)5-s + (0.540 − 0.158i)7-s + (0.533 − 1.16i)9-s + (−1.27 − 0.817i)15-s + (0.557 − 0.643i)21-s + (−0.755 − 0.654i)23-s + (−0.654 + 0.755i)25-s + (−0.0612 − 0.425i)27-s + (−0.118 + 0.822i)29-s + (−0.368 − 0.425i)35-s + (0.797 + 1.74i)41-s + (1.66 − 1.07i)43-s − 1.28·45-s − 1.81·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.130 + 0.991i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :0),\ 0.130 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.678150349\)
\(L(\frac12)\) \(\approx\) \(1.678150349\)
\(L(1)\) 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 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (0.755 + 0.654i)T \)
good3 \( 1 + (-1.27 + 0.817i)T + (0.415 - 0.909i)T^{2} \)
7 \( 1 + (-0.540 + 0.158i)T + (0.841 - 0.540i)T^{2} \)
11 \( 1 + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (-0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.415 - 0.909i)T^{2} \)
37 \( 1 + (0.654 + 0.755i)T^{2} \)
41 \( 1 + (-0.797 - 1.74i)T + (-0.654 + 0.755i)T^{2} \)
43 \( 1 + (-1.66 + 1.07i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + 1.81T + T^{2} \)
53 \( 1 + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \)
67 \( 1 + (-0.708 + 0.817i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (-0.841 - 0.540i)T^{2} \)
83 \( 1 + (-0.449 + 0.983i)T + (-0.654 - 0.755i)T^{2} \)
89 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (0.654 - 0.755i)T^{2} \)
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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

−9.043883938321858802765413044862, −8.288930650256984835444554117776, −7.934748929374722267046294856215, −7.21596307700679144244450240558, −6.21528557187779552835220347364, −5.03911834197044685804610161347, −4.21366389679827176302665879657, −3.25369728798643984754011083655, −2.14372405106164536355858951246, −1.21124367485581951808216410072, 2.05851927808883723713675847581, 2.86633313707992347076146144957, 3.78089161477507288015769524297, 4.32540065371331391599371973365, 5.50965618437103400294812187150, 6.55039614734892784265011767547, 7.66265062139495134630054744531, 7.980596711122766750876182091820, 8.841132390356708905920538157919, 9.645585811059878048730036040836

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