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Properties

Label 1-1840-1840.469-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.0126 + 0.999i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (−0.989 − 0.142i)3-s + (−0.654 − 0.755i)7-s + (0.959 + 0.281i)9-s + (−0.540 + 0.841i)11-s + (−0.755 − 0.654i)13-s + (−0.415 − 0.909i)17-s + (−0.909 − 0.415i)19-s + (0.540 + 0.841i)21-s + (−0.909 − 0.415i)27-s + (0.909 − 0.415i)29-s + (−0.142 − 0.989i)31-s + (0.654 − 0.755i)33-s + (−0.281 + 0.959i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)3-s + (−0.654 − 0.755i)7-s + (0.959 + 0.281i)9-s + (−0.540 + 0.841i)11-s + (−0.755 − 0.654i)13-s + (−0.415 − 0.909i)17-s + (−0.909 − 0.415i)19-s + (0.540 + 0.841i)21-s + (−0.909 − 0.415i)27-s + (0.909 − 0.415i)29-s + (−0.142 − 0.989i)31-s + (0.654 − 0.755i)33-s + (−0.281 + 0.959i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0126 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0126 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.0126 + 0.999i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.0126 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1581099363 + 0.1601194477i\)
\(L(\frac12)\) \(\approx\) \(0.1581099363 + 0.1601194477i\)
\(L(1)\) \(\approx\) \(0.5614130207 - 0.08432391315i\)
\(L(1)\) \(\approx\) \(0.5614130207 - 0.08432391315i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (-0.540 + 0.841i)T \)
13 \( 1 + (-0.755 - 0.654i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (-0.909 - 0.415i)T \)
29 \( 1 + (0.909 - 0.415i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (-0.281 + 0.959i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.755 + 0.654i)T \)
59 \( 1 + (-0.755 - 0.654i)T \)
61 \( 1 + (-0.989 + 0.142i)T \)
67 \( 1 + (0.540 + 0.841i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (0.281 - 0.959i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.57263442938077593408728075566, −19.28830859905234361069115204687, −18.44283718298616530272937152163, −17.744177915940132961144728845732, −16.97222803422263137523234351787, −16.229004723608899105098658231071, −15.81161811600876133024905629583, −14.92302502826661101846952279006, −14.08950346621484314256196372587, −12.899863366289488015043168261760, −12.562668355947616211516196768230, −11.82891884946590907995763902193, −10.85027487494716366458086935323, −10.4439854823523600467226649995, −9.4279282615418102478874613466, −8.7532097186296606780140846959, −7.761257319170070270471905375904, −6.62121758801517106267130043805, −6.2155997087519089699562917646, −5.39753364271963057974723234138, −4.588357507769173552656817748187, −3.654259488598226016138589715377, −2.5977066032282415411576799020, −1.59789381963375422413664522011, −0.12236256504045412629661174464, 0.78752999625438955882527659259, 2.137543181916576857635726211673, 3.01889097572413060540294962056, 4.48836852844502290230016388941, 4.64260941739070976609422006433, 5.82870989811394111497790587728, 6.59033568951419003606197421735, 7.31205468593395761937149078980, 7.864714602805974686136738405840, 9.303419472609766221238136105820, 10.00047558880444995738147094270, 10.53817906719974806197376419013, 11.3304896302793498160890138157, 12.23976835627652758517533573290, 12.880755473886537516430915590036, 13.36194617214933301081389752048, 14.41642871499453065066895463501, 15.52992056077043711227030511781, 15.81355512799007875440008996832, 16.8895906794478267249213611151, 17.33808648668223845235584302866, 17.93984360630975968062292516250, 18.818346643373526082940222213570, 19.56826068807539006079525789216, 20.3091367657615353309077933722

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