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Properties

Label 1-1412-1412.1123-r1-0-0
Degree $1$
Conductor $1412$
Sign $-0.999 + 0.0219i$
Analytic cond. $151.740$
Root an. cond. $151.740$
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  + (−0.281 + 0.959i)3-s + (0.281 − 0.959i)5-s − i·7-s + (−0.841 − 0.540i)9-s + (0.142 + 0.989i)11-s + (0.909 + 0.415i)13-s + (0.841 + 0.540i)15-s + (−0.654 + 0.755i)17-s + (0.415 + 0.909i)19-s + (−0.959 − 0.281i)21-s + (−0.142 + 0.989i)23-s + (−0.841 − 0.540i)25-s + (0.755 − 0.654i)27-s + (0.841 + 0.540i)29-s + (−0.989 + 0.142i)31-s + ⋯
L(s)  = 1  + (−0.281 + 0.959i)3-s + (0.281 − 0.959i)5-s − i·7-s + (−0.841 − 0.540i)9-s + (0.142 + 0.989i)11-s + (0.909 + 0.415i)13-s + (0.841 + 0.540i)15-s + (−0.654 + 0.755i)17-s + (0.415 + 0.909i)19-s + (−0.959 − 0.281i)21-s + (−0.142 + 0.989i)23-s + (−0.841 − 0.540i)25-s + (0.755 − 0.654i)27-s + (0.841 + 0.540i)29-s + (−0.989 + 0.142i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $-0.999 + 0.0219i$
Analytic conductor: \(151.740\)
Root analytic conductor: \(151.740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (1123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1412,\ (1:\ ),\ -0.999 + 0.0219i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01717554473 + 1.561341871i\)
\(L(\frac12)\) \(\approx\) \(0.01717554473 + 1.561341871i\)
\(L(1)\) \(\approx\) \(0.8658191717 + 0.5067559076i\)
\(L(1)\) \(\approx\) \(0.8658191717 + 0.5067559076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
353 \( 1 \)
good3 \( 1 + (-0.281 + 0.959i)T \)
5 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.142 + 0.989i)T \)
13 \( 1 + (0.909 + 0.415i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (-0.989 + 0.142i)T \)
37 \( 1 + (0.755 + 0.654i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (-0.281 + 0.959i)T \)
59 \( 1 - iT \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.540 + 0.841i)T \)
73 \( 1 + (0.841 - 0.540i)T \)
79 \( 1 + (-0.281 - 0.959i)T \)
83 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (0.755 - 0.654i)T \)
97 \( 1 + (-0.654 - 0.755i)T \)
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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

−20.05181869934552433138955059809, −19.36515609220152606849254043716, −18.60927073486219056367126570840, −17.93306335590528624375680264263, −17.50365257502531614981304422534, −16.45764887516384855986488955204, −15.84447183654706544954098021462, −14.57434627556553496995027155176, −13.89308669713688160869281019916, −13.505535685333933800046133542065, −12.75001820124297710137912325517, −11.404603862007581846910345560441, −11.12225158257113122693720375654, −10.46325075276325218419608260158, −9.27714554305163298543472978834, −8.29139880887401128161219435945, −7.49717033043867923713372664384, −6.72598545927390326136190273866, −6.247625216950699199959245219275, −5.30406606089405162236757462067, −4.02319922404154961695555784864, −3.03795691011846211540524351121, −2.3054277171130446986339339623, −0.97541287416683448323307343990, −0.35942246569977944805354541795, 1.28509596283273626506548800547, 2.12685714675941397702949778651, 3.47931770306992494833414275214, 4.26982048651760672808434902919, 5.076254244799246428337242313392, 5.77778126242836974082471427941, 6.47117265959629628530709992324, 7.95724062752444037685059373935, 8.78515201830708863899290877521, 9.28389206420320816819978457614, 9.98595747164635822259450345112, 10.93667811790922533463052957837, 11.88455375497341339045449784121, 12.3341536337436134381772985037, 13.27298400125947249472365577743, 14.24053646828314658623461108590, 15.15673914918892987863580533853, 15.67176198587992930022931665569, 16.3744925595951321835554624256, 17.09364375162240589528048573082, 17.86153402221193425516007061144, 18.50892712389233882693995189621, 19.82470730918804911912212681286, 20.25826711444692880314022559512, 21.08189609491050098920636371400

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