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Properties

Label 1-247-247.12-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: $\chi_{247} (12, \cdot )$
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

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

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