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Properties

Label 1-247-247.103-r1-0-0
Degree $1$
Conductor $247$
Sign $-0.813 - 0.582i$
Analytic cond. $26.5438$
Root an. cond. $26.5438$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s − 7-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)10-s − 11-s − 12-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s − 7-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)10-s − 11-s − 12-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(247\)    =    \(13 \cdot 19\)
Sign: $-0.813 - 0.582i$
Analytic conductor: \(26.5438\)
Root analytic conductor: \(26.5438\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{247} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 247,\ (1:\ ),\ -0.813 - 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009828437007 + 0.03060390722i\)
\(L(\frac12)\) \(\approx\) \(0.009828437007 + 0.03060390722i\)
\(L(1)\) \(\approx\) \(0.6891405191 + 0.06767788018i\)
\(L(1)\) \(\approx\) \(0.6891405191 + 0.06767788018i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−26.000882944178026547699755584906, −25.334277598617775090093176201271, −24.52008049450574544977421992639, −23.77868941262223789026359467924, −23.01081358923681954690630379125, −21.64407324644112489217271801147, −20.251742745259532491949800544096, −19.65810934883570145197049460788, −18.60566636415421807963467269124, −17.89122912864114549606058867817, −16.883947715067255248021558897561, −16.03852138858030830983350383258, −15.038332522636545346561215838413, −13.79926899082053761903932990772, −13.1389815589200180540359934556, −12.41271951335125476519835445186, −10.47183025065657975848968348353, −9.49709745813123576120432524274, −8.58274277683328361223018922419, −7.8504225845657938213056836246, −6.52294445018728074461697059916, −5.92956621631004524409662052816, −4.514614536286377570843114864222, −2.65232550302550611231049021595, −1.2556626024540975632942661001, 0.01156185522804334774354710382, 2.35187447556834674256835308090, 2.935860843582099180816546975260, 4.0096750632272249714731861419, 5.451959069218245281652909395324, 7.04269067803781639264968063458, 8.222502676990758093692531986328, 9.46505388649757119749928987863, 9.99783077700618920989518186636, 10.736200218377580640867824736398, 11.839474586882663763572554758881, 13.45096769199532068961485744788, 13.6965225750952231337518044671, 15.305104581583048551637524692517, 16.057156965368427851975061358614, 17.20802054579157796037800504408, 18.26843702950235029435020489389, 19.09268701443065475182605500163, 19.94008995160704059281816282656, 20.889214554359514551948368567104, 21.674878312455731756378581283329, 22.3906097872564918386494375922, 23.138815771964745544043436754856, 25.20744256790742753584739546174, 25.77062019940736163284538815102

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