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Properties

Label 1-247-247.158-r1-0-0
Degree $1$
Conductor $247$
Sign $-0.610 + 0.791i$
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  + (0.642 − 0.766i)2-s + (−0.939 + 0.342i)3-s + (−0.173 − 0.984i)4-s + (0.342 + 0.939i)5-s + (−0.342 + 0.939i)6-s + (−0.866 − 0.5i)7-s + (−0.866 − 0.5i)8-s + (0.766 − 0.642i)9-s + (0.939 + 0.342i)10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)12-s + (−0.939 + 0.342i)14-s + (−0.642 − 0.766i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s − i·18-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + (−0.939 + 0.342i)3-s + (−0.173 − 0.984i)4-s + (0.342 + 0.939i)5-s + (−0.342 + 0.939i)6-s + (−0.866 − 0.5i)7-s + (−0.866 − 0.5i)8-s + (0.766 − 0.642i)9-s + (0.939 + 0.342i)10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)12-s + (−0.939 + 0.342i)14-s + (−0.642 − 0.766i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s − i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(247\)    =    \(13 \cdot 19\)
Sign: $-0.610 + 0.791i$
Analytic conductor: \(26.5438\)
Root analytic conductor: \(26.5438\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{247} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 247,\ (1:\ ),\ -0.610 + 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006553099223 + 0.01333355437i\)
\(L(\frac12)\) \(\approx\) \(0.006553099223 + 0.01333355437i\)
\(L(1)\) \(\approx\) \(0.8081896171 - 0.2726387535i\)
\(L(1)\) \(\approx\) \(0.8081896171 - 0.2726387535i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.642 - 0.766i)T \)
3 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (0.342 + 0.939i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.342 + 0.939i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.984 + 0.173i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (0.642 - 0.766i)T \)
73 \( 1 + (-0.642 + 0.766i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.984 + 0.173i)T \)
97 \( 1 + (0.342 + 0.939i)T \)
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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

−25.26441708066328730676687370978, −24.50447300717595139576626213334, −23.79249928748319654901531765493, −22.742530603641812048683874016634, −22.18121256960865014818701163799, −21.29443280458191560239396565848, −20.11598748960180377478689609025, −18.77539960250741165438491963160, −17.73982053079982248623456523063, −16.74972161622700892154251492526, −16.4532332063156682539194614182, −15.342677884341117233161043270198, −14.09595428545981709596015047399, −12.89808270246136121046598429591, −12.4812161732039468650031901626, −11.69063691369233974861040132946, −9.93971255117473135958474534549, −8.94029084394068632387881459582, −7.660577860009897659451945599, −6.47684878499767954033828642452, −5.81189834359732292067283689375, −4.88008965586910464670075633665, −3.70715967705707524408034177585, −1.80513949023568802938946449039, −0.004268029338402851994130500761, 1.49522361180426454088411281186, 3.2823761699320175265314492679, 3.87566550235728002573711520481, 5.451376404392459250894108816848, 6.22827441161071682384065217061, 7.094460264478832863116089450956, 9.479454775019771240286892238618, 10.00208692764602361479376763444, 10.98477121247998307059987199597, 11.70432951389943196118535446684, 12.76398015320362668751576336106, 13.833975610509011450948665927420, 14.66248576964650877189543822123, 15.82878770993961444776416203979, 16.80253369576533357226844535115, 17.963609193057949948622662720297, 18.86320475207528788562792550460, 19.6784549735256334994051569470, 20.93191205300356792622964955122, 21.866007111560170773466447018055, 22.39216504548663571597789507239, 23.07885010084474664347007531712, 23.88208410863505512389722660574, 25.189581447221254285784644130246, 26.42630119535879794386295544981

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