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Properties

Label 1-1412-1412.191-r1-0-0
Degree $1$
Conductor $1412$
Sign $-0.893 - 0.449i$
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  + (−0.989 + 0.142i)3-s + (0.989 − 0.142i)5-s − i·7-s + (0.959 − 0.281i)9-s + (0.654 + 0.755i)11-s + (−0.540 + 0.841i)13-s + (−0.959 + 0.281i)15-s + (0.415 − 0.909i)17-s + (0.841 − 0.540i)19-s + (−0.142 − 0.989i)21-s + (−0.654 + 0.755i)23-s + (0.959 − 0.281i)25-s + (−0.909 + 0.415i)27-s + (−0.959 + 0.281i)29-s + (−0.755 + 0.654i)31-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)3-s + (0.989 − 0.142i)5-s − i·7-s + (0.959 − 0.281i)9-s + (0.654 + 0.755i)11-s + (−0.540 + 0.841i)13-s + (−0.959 + 0.281i)15-s + (0.415 − 0.909i)17-s + (0.841 − 0.540i)19-s + (−0.142 − 0.989i)21-s + (−0.654 + 0.755i)23-s + (0.959 − 0.281i)25-s + (−0.909 + 0.415i)27-s + (−0.959 + 0.281i)29-s + (−0.755 + 0.654i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $-0.893 - 0.449i$
Analytic conductor: \(151.740\)
Root analytic conductor: \(151.740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1412,\ (1:\ ),\ -0.893 - 0.449i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08036058357 + 0.3388303744i\)
\(L(\frac12)\) \(\approx\) \(-0.08036058357 + 0.3388303744i\)
\(L(1)\) \(\approx\) \(0.8154809454 + 0.2035571665i\)
\(L(1)\) \(\approx\) \(0.8154809454 + 0.2035571665i\)

Euler product

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

−20.30411990322594485598367433078, −19.2510692230788353900767902248, −18.53076519198149009917436568307, −17.69421529213118116053972928337, −17.10378476372690842367881068769, −16.70009607756885424706788174076, −15.876619883474220978935306362038, −14.55365735103247789075712280515, −14.13870997556533669335661050128, −13.04774657936135732936952762698, −12.708733158028561287626138530, −11.53599533004990898555361388925, −10.88128337839261625671270054044, −10.04443656729212491851895373159, −9.73697254962765312292754114744, −8.297653634828994315414050358358, −7.436366945843115020564514147416, −6.58125404846806967199660389751, −5.86028155776598402841234555835, −5.28789516183867505844561334469, −4.13900993304733991668042334712, −3.28506244116088788601060006582, −1.83004959461048129385048484458, −1.09676496427836119632088034708, −0.07688819416020337535873592640, 1.45011550121499054484591409039, 1.99867881554500188692056148287, 3.28419677432430438793877774053, 4.60825916871387634681217915094, 5.23904413008849232467691299059, 5.83603019548706903662368355037, 6.80542063025285853654839960648, 7.37004581092904682760108049673, 8.99808130870838331221811888213, 9.48189002209155693514029971789, 9.97977400244555195255501289507, 11.195758869194341673813333933600, 11.8890730686253913702613837533, 12.362687530545415844779388394058, 13.29416052585235091710698385270, 14.23358367296377106986078707278, 14.93639716165315493541330760745, 15.95831127272728613599947870773, 16.500624621772018548696874183225, 17.34482815495968408873039704404, 17.99205064893594452106504934517, 18.40467084875584884660263330965, 19.43646144156934580378997665571, 20.437860612560584457253885860324, 21.26196605108508184022350672849

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