Properties

Label 1-47e2-2209.31-r1-0-0
Degree $1$
Conductor $2209$
Sign $-0.889 - 0.456i$
Analytic cond. $237.390$
Root an. cond. $237.390$
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  + (−0.811 − 0.584i)2-s + (−0.846 + 0.531i)3-s + (0.315 + 0.948i)4-s + (0.701 − 0.712i)5-s + (0.997 + 0.0638i)6-s + (0.999 + 0.0116i)7-s + (0.299 − 0.954i)8-s + (0.434 − 0.900i)9-s + (−0.985 + 0.167i)10-s + (0.160 − 0.987i)11-s + (−0.772 − 0.635i)12-s + (0.973 + 0.227i)13-s + (−0.804 − 0.594i)14-s + (−0.214 + 0.976i)15-s + (−0.800 + 0.599i)16-s + (−0.998 + 0.0609i)17-s + ⋯
L(s)  = 1  + (−0.811 − 0.584i)2-s + (−0.846 + 0.531i)3-s + (0.315 + 0.948i)4-s + (0.701 − 0.712i)5-s + (0.997 + 0.0638i)6-s + (0.999 + 0.0116i)7-s + (0.299 − 0.954i)8-s + (0.434 − 0.900i)9-s + (−0.985 + 0.167i)10-s + (0.160 − 0.987i)11-s + (−0.772 − 0.635i)12-s + (0.973 + 0.227i)13-s + (−0.804 − 0.594i)14-s + (−0.214 + 0.976i)15-s + (−0.800 + 0.599i)16-s + (−0.998 + 0.0609i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2209\)    =    \(47^{2}\)
Sign: $-0.889 - 0.456i$
Analytic conductor: \(237.390\)
Root analytic conductor: \(237.390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2209} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2209,\ (1:\ ),\ -0.889 - 0.456i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2319797965 - 0.9591493326i\)
\(L(\frac12)\) \(\approx\) \(0.2319797965 - 0.9591493326i\)
\(L(1)\) \(\approx\) \(0.6732216621 - 0.2463888092i\)
\(L(1)\) \(\approx\) \(0.6732216621 - 0.2463888092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (-0.811 - 0.584i)T \)
3 \( 1 + (-0.846 + 0.531i)T \)
5 \( 1 + (0.701 - 0.712i)T \)
7 \( 1 + (0.999 + 0.0116i)T \)
11 \( 1 + (0.160 - 0.987i)T \)
13 \( 1 + (0.973 + 0.227i)T \)
17 \( 1 + (-0.998 + 0.0609i)T \)
19 \( 1 + (-0.0421 - 0.999i)T \)
23 \( 1 + (0.837 + 0.546i)T \)
29 \( 1 + (-0.908 + 0.416i)T \)
31 \( 1 + (-0.984 + 0.173i)T \)
37 \( 1 + (-0.262 + 0.964i)T \)
41 \( 1 + (0.965 + 0.261i)T \)
43 \( 1 + (0.999 - 0.0203i)T \)
53 \( 1 + (-0.990 + 0.136i)T \)
59 \( 1 + (0.555 - 0.831i)T \)
61 \( 1 + (-0.994 - 0.101i)T \)
67 \( 1 + (-0.854 + 0.519i)T \)
71 \( 1 + (0.460 - 0.887i)T \)
73 \( 1 + (-0.634 + 0.772i)T \)
79 \( 1 + (0.248 + 0.968i)T \)
83 \( 1 + (0.0943 - 0.995i)T \)
89 \( 1 + (-0.586 - 0.810i)T \)
97 \( 1 + (-0.206 + 0.978i)T \)
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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

−19.56040507423605417107853270348, −18.70344874130855018216314245221, −18.17783261050598322269384237864, −17.782516863193236102908685981808, −17.22958774093465776334677020862, −16.519220935477838777886354876061, −15.58539544864577625407954864920, −14.83580510560575369673103683076, −14.252724856466122516784522136136, −13.40238025091742807947439896157, −12.560834389788716561182019338, −11.45752019975494751252051558796, −10.82219982472142070644224363938, −10.58560780281867450944967224456, −9.46987290653715445403983942509, −8.74512210253078483687652488144, −7.57912810019364197074022867226, −7.337830552712834966681046257397, −6.34268274186689968707378277147, −5.8464044177574658169727628807, −5.08070654945762607219526224194, −4.12346094682936019550820456679, −2.312098305854403352675077818961, −1.82414043805819502750767082972, −1.03998246274267251005755195296, 0.27680266735378952421373001439, 1.166176472115567586176421283191, 1.72746465070695414919482106975, 3.01328468225365865995826648711, 4.05461177645673606086695447376, 4.75344022119580288537876372237, 5.63649027058525448481821072450, 6.41670835961323916269852522517, 7.3532202822821557977524628922, 8.55978185350105676781961101460, 8.95244010037623838346919486815, 9.51171276760259588043249233427, 10.70515377217385583260738442802, 11.15499317871345534805164261962, 11.42914126554470941599654834965, 12.58783009052076314891191266573, 13.202449142290201918127381044798, 13.96555934546445900801856065686, 15.21539808400694563188114922777, 15.947576333486223801552756117642, 16.58967214076746332084972733729, 17.22271375009065592365189135735, 17.751343310131759943943807436174, 18.2706571602499691954706591250, 19.13392172919083067844559823392

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