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Properties

Label 1-337-337.64-r0-0-0
Degree $1$
Conductor $337$
Sign $-0.821 + 0.570i$
Analytic cond. $1.56502$
Root an. cond. $1.56502$
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  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.222 + 0.974i)7-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + (−0.222 + 0.974i)11-s + (0.623 + 0.781i)12-s + (0.623 − 0.781i)13-s + (−0.900 − 0.433i)14-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + (0.623 − 0.781i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.222 + 0.974i)7-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + (−0.222 + 0.974i)11-s + (0.623 + 0.781i)12-s + (0.623 − 0.781i)13-s + (−0.900 − 0.433i)14-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + (0.623 − 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 337 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 337 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(337\)
Sign: $-0.821 + 0.570i$
Analytic conductor: \(1.56502\)
Root analytic conductor: \(1.56502\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{337} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 337,\ (0:\ ),\ -0.821 + 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2242904225 + 0.7154212506i\)
\(L(\frac12)\) \(\approx\) \(0.2242904225 + 0.7154212506i\)
\(L(1)\) \(\approx\) \(0.5729774080 + 0.4511020831i\)
\(L(1)\) \(\approx\) \(0.5729774080 + 0.4511020831i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad337 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 + (-0.222 - 0.974i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (-0.222 + 0.974i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−24.31263689460030622992911683035, −23.46732812847673697740199702179, −22.78084565206500216958968811544, −21.63631501258259134636026260168, −21.05576203293492097230067326202, −20.54099557466588736341238885012, −19.18652561688749871260936648427, −18.4625750254415708421478185089, −17.16493152557040694968854399468, −16.87631812467898705861895062588, −16.109887601232265276252742746000, −14.29096529916478126879243342364, −13.41719898272600615964184601189, −12.626455663725410163427526371890, −11.67243845894514085916904357838, −10.63201262027845505718936469637, −10.14820636547818562146551780136, −9.092291219016590791926349474217, −8.11848522833692138949486205316, −6.41682490859998415078275802971, −5.442031577284917543827487084323, −4.29247475608773532356673266078, −3.57649551262546799157210949280, −1.67195186232204587274493812955, −0.63705047258646146155226913208, 1.4681185611641498768102525171, 3.01374870432662791334052493033, 5.06098134779144743979891350985, 5.46880936735871994069723173088, 6.66687093247694940873445057143, 7.10413994300598936043891486131, 8.42867922866593161235348999327, 9.672697352908581699132136174758, 10.38688175435036739051605039271, 11.580501512357155574732655515887, 12.82567002913133416309112113793, 13.45044101764070478107100972105, 14.73224739116680852451061134352, 15.49194716771230928830663771139, 16.31129485669384416726102134321, 17.6101904272912509087424916608, 17.933890591373497522242605568767, 18.575504186244028989334715322639, 19.56013513567255952032920383938, 21.33417284501262562942167733110, 22.09527704407362793835575980639, 22.9422605100402176442344569055, 23.33441226107782361389125159418, 24.63786122313518882476644690983, 25.418934548201912682889010377263

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