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Properties

Label 1-1832-1832.1323-r1-0-0
Degree $1$
Conductor $1832$
Sign $-0.705 - 0.708i$
Analytic cond. $196.875$
Root an. cond. $196.875$
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  + (−0.998 + 0.0550i)3-s + (−0.350 − 0.936i)5-s + (0.137 − 0.990i)7-s + (0.993 − 0.110i)9-s + (−0.879 + 0.475i)11-s + (−0.986 − 0.164i)13-s + (0.401 + 0.915i)15-s + (−0.986 − 0.164i)17-s + (0.635 + 0.771i)19-s + (−0.0825 + 0.996i)21-s + (0.716 + 0.697i)23-s + (−0.754 + 0.656i)25-s + (−0.986 + 0.164i)27-s + (−0.926 + 0.376i)29-s + (0.851 + 0.523i)31-s + ⋯
L(s)  = 1  + (−0.998 + 0.0550i)3-s + (−0.350 − 0.936i)5-s + (0.137 − 0.990i)7-s + (0.993 − 0.110i)9-s + (−0.879 + 0.475i)11-s + (−0.986 − 0.164i)13-s + (0.401 + 0.915i)15-s + (−0.986 − 0.164i)17-s + (0.635 + 0.771i)19-s + (−0.0825 + 0.996i)21-s + (0.716 + 0.697i)23-s + (−0.754 + 0.656i)25-s + (−0.986 + 0.164i)27-s + (−0.926 + 0.376i)29-s + (0.851 + 0.523i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1832\)    =    \(2^{3} \cdot 229\)
Sign: $-0.705 - 0.708i$
Analytic conductor: \(196.875\)
Root analytic conductor: \(196.875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1832} (1323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1832,\ (1:\ ),\ -0.705 - 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2498133781 - 0.6017205949i\)
\(L(\frac12)\) \(\approx\) \(0.2498133781 - 0.6017205949i\)
\(L(1)\) \(\approx\) \(0.6248505143 - 0.1567749050i\)
\(L(1)\) \(\approx\) \(0.6248505143 - 0.1567749050i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.998 + 0.0550i)T \)
5 \( 1 + (-0.350 - 0.936i)T \)
7 \( 1 + (0.137 - 0.990i)T \)
11 \( 1 + (-0.879 + 0.475i)T \)
13 \( 1 + (-0.986 - 0.164i)T \)
17 \( 1 + (-0.986 - 0.164i)T \)
19 \( 1 + (0.635 + 0.771i)T \)
23 \( 1 + (0.716 + 0.697i)T \)
29 \( 1 + (-0.926 + 0.376i)T \)
31 \( 1 + (0.851 + 0.523i)T \)
37 \( 1 + (0.821 + 0.569i)T \)
41 \( 1 + (0.592 - 0.805i)T \)
43 \( 1 + (-0.0825 + 0.996i)T \)
47 \( 1 + (0.993 - 0.110i)T \)
53 \( 1 + (-0.546 - 0.837i)T \)
59 \( 1 + (0.821 - 0.569i)T \)
61 \( 1 + (0.401 - 0.915i)T \)
67 \( 1 + (0.592 + 0.805i)T \)
71 \( 1 + (-0.0275 - 0.999i)T \)
73 \( 1 + (0.754 - 0.656i)T \)
79 \( 1 + (-0.926 - 0.376i)T \)
83 \( 1 + (0.904 + 0.426i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.191 + 0.981i)T \)
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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

−20.21050910227187426845852975084, −19.11960363903077356181045074816, −18.768143569227271448521243192094, −18.06255115015304308440048588986, −17.48054310839296232262641517430, −16.58957917755766469254934602599, −15.59366194879479751557326571499, −15.38966346082758393454174481477, −14.512591995982602497503439187041, −13.40552894537392374477764664395, −12.73062056373629108890808020995, −11.8414349272426576249114756521, −11.29172974920634052290551197048, −10.75104658272554478247293840653, −9.85792404716094842820860424682, −9.01402001566730349828922740003, −7.88258386245784134906918010395, −7.213640538227811841667390843104, −6.42362966908712820616963705314, −5.66712859167291407208715968513, −4.92083696613934672897006586743, −4.07146613371809821940941605972, −2.62068697741028626436528626040, −2.38361312880524888581483063423, −0.67892777668388488206540426307, 0.22678730544921009523830296629, 1.00672490531957917954495568774, 2.00538380770150030375207919854, 3.46399956752661789404711446208, 4.46693010807074170559909714263, 4.91493501149430131676954185849, 5.58899711192273681187967771761, 6.81133426058031349821922819337, 7.47765773355661146420626532014, 8.048374136071726583677783117, 9.39642474175339670549489698837, 9.91905710068962735042547371530, 10.79912445032422172321579698796, 11.45597152725216591336636588896, 12.2659918950418911129431344813, 12.95463662040619225074324043365, 13.43252577168797067743492490393, 14.62257570095521419090425968059, 15.573334992344967182826585196462, 16.07566499128283160944951843382, 16.862509356530240037777680524142, 17.39873305193283346227856509436, 17.93956531418267687178417774498, 18.9744337509494787392562492219, 19.79563257930807341496816134463

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