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Properties

Label 1-2669-2669.1015-r0-0-0
Degree $1$
Conductor $2669$
Sign $0.584 + 0.811i$
Analytic cond. $12.3947$
Root an. cond. $12.3947$
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  + (−0.0603 − 0.998i)2-s + (−0.860 + 0.508i)3-s + (−0.992 + 0.120i)4-s + (0.525 − 0.850i)5-s + (0.559 + 0.828i)6-s + (0.963 − 0.268i)7-s + (0.180 + 0.983i)8-s + (0.482 − 0.875i)9-s + (−0.880 − 0.473i)10-s + (−0.828 + 0.559i)11-s + (0.793 − 0.608i)12-s + (−0.866 + 0.5i)13-s + (−0.326 − 0.945i)14-s + (−0.0201 + 0.999i)15-s + (0.970 − 0.239i)16-s + ⋯
L(s)  = 1  + (−0.0603 − 0.998i)2-s + (−0.860 + 0.508i)3-s + (−0.992 + 0.120i)4-s + (0.525 − 0.850i)5-s + (0.559 + 0.828i)6-s + (0.963 − 0.268i)7-s + (0.180 + 0.983i)8-s + (0.482 − 0.875i)9-s + (−0.880 − 0.473i)10-s + (−0.828 + 0.559i)11-s + (0.793 − 0.608i)12-s + (−0.866 + 0.5i)13-s + (−0.326 − 0.945i)14-s + (−0.0201 + 0.999i)15-s + (0.970 − 0.239i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2669\)    =    \(17 \cdot 157\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(12.3947\)
Root analytic conductor: \(12.3947\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2669} (1015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2669,\ (0:\ ),\ 0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3448363274 + 0.1765305124i\)
\(L(\frac12)\) \(\approx\) \(0.3448363274 + 0.1765305124i\)
\(L(1)\) \(\approx\) \(0.6166013725 - 0.2717116043i\)
\(L(1)\) \(\approx\) \(0.6166013725 - 0.2717116043i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
157 \( 1 \)
good2 \( 1 + (-0.0603 - 0.998i)T \)
3 \( 1 + (-0.860 + 0.508i)T \)
5 \( 1 + (0.525 - 0.850i)T \)
7 \( 1 + (0.963 - 0.268i)T \)
11 \( 1 + (-0.828 + 0.559i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.941 - 0.335i)T \)
23 \( 1 + (0.805 - 0.592i)T \)
29 \( 1 + (-0.728 - 0.685i)T \)
31 \( 1 + (-0.817 + 0.576i)T \)
37 \( 1 + (0.229 - 0.973i)T \)
41 \( 1 + (0.150 + 0.988i)T \)
43 \( 1 + (-0.140 + 0.990i)T \)
47 \( 1 + (-0.774 - 0.632i)T \)
53 \( 1 + (0.647 + 0.761i)T \)
59 \( 1 + (0.983 + 0.180i)T \)
61 \( 1 + (-0.670 + 0.741i)T \)
67 \( 1 + (-0.568 + 0.822i)T \)
71 \( 1 + (-0.455 + 0.890i)T \)
73 \( 1 + (0.860 - 0.508i)T \)
79 \( 1 + (-0.437 + 0.899i)T \)
83 \( 1 + (-0.219 - 0.975i)T \)
89 \( 1 + (0.999 + 0.0402i)T \)
97 \( 1 + (0.473 + 0.880i)T \)
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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

−18.655555637793448751103201292090, −18.48137666421480971194279287845, −17.65481801028847491274060893277, −17.19678575601486339189618294542, −16.63609890870203671854131031407, −15.584165408171471392414083244069, −14.95166067961989621695668977120, −14.43355930017259957939740223426, −13.48421535143142655761030602348, −13.03345743054659747183054788690, −12.16120450676202694068604877191, −11.098760762271283383326423292599, −10.661533574751493503300135639927, −9.90339318476346512949241770101, −8.87642205929130751856590102301, −7.91466636994251364434447519095, −7.48326874518697872607780428750, −6.74640923403911865307107488603, −5.89729902213086578976760684181, −5.33500827085193441121666840020, −4.88286677200680413883952223743, −3.59588435998373127523273381831, −2.39492444428422785753024355064, −1.54449332372211411399090196228, −0.16060575933246361783882039365, 0.979213661747786910979176636334, 1.86917525904769660536694443280, 2.593075674432438911069309696178, 4.08719988767518983426760983962, 4.566167835707105564143375535779, 5.08048948996684543128472782564, 5.73163532050204881869144260718, 7.00865023244213459520717138920, 7.94850868821762419441664534051, 8.868042277842891707639555381665, 9.4573925175399573875348996416, 10.22999016810226877451692249053, 10.76219720221417994325801492131, 11.50374413206452023972486478920, 12.16915389458999426360698452966, 12.915388423771967605239932029173, 13.30138837636047771541440338034, 14.6675054281587721894768806638, 14.82093570238907447678411053533, 16.20109553759485574532853474102, 16.86795254266683478684753963785, 17.36426232799478034192678996412, 17.941263198925723205390552866251, 18.52822141642465082315136949511, 19.63697494882561354481293663602

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