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Properties

Label 1-1287-1287.1271-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.350 - 0.936i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (0.913 − 0.406i)5-s + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (0.5 − 0.866i)10-s + (0.978 + 0.207i)14-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (0.669 − 0.743i)19-s + (−0.104 − 0.994i)20-s + (0.5 + 0.866i)23-s + (0.669 − 0.743i)25-s + (0.913 − 0.406i)28-s + (0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (0.913 − 0.406i)5-s + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (0.5 − 0.866i)10-s + (0.978 + 0.207i)14-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (0.669 − 0.743i)19-s + (−0.104 − 0.994i)20-s + (0.5 + 0.866i)23-s + (0.669 − 0.743i)25-s + (0.913 − 0.406i)28-s + (0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.350 - 0.936i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.350 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.713279747 - 1.881394364i\)
\(L(\frac12)\) \(\approx\) \(2.713279747 - 1.881394364i\)
\(L(1)\) \(\approx\) \(1.894938260 - 0.8353868985i\)
\(L(1)\) \(\approx\) \(1.894938260 - 0.8353868985i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.669 - 0.743i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−20.98770093309680930343944699891, −20.82240828516715309510027030302, −19.97415034298345134925924454214, −18.48287154595961895080182869460, −18.08034746712589602750720732677, −17.03456606065183258809018969717, −16.75593887471647659844377038960, −15.695296276781434724188334325959, −14.81487462305885928042359944014, −14.02584396014545173056329932133, −13.84339166773527677365899551043, −12.87214312709327357786067655345, −11.97133458007159769077984654211, −11.08723958567367670132015017202, −10.37333648751578908833711556566, −9.33246473074034738369373207929, −8.361048685782582924768001704542, −7.38114944706867255417775578440, −6.873064668032176057021855283499, −5.89877686954215899793652328129, −5.12049845751279262668458011939, −4.38382180947801998976148119509, −3.27319527217423297944741087808, −2.460264887052423052211580455, −1.30952778349657560043719298829, 1.14067530937301397930546630731, 1.93963868970108253645114052897, 2.70008426864851755377179116784, 3.81997177366749094102678562259, 4.93312261578275452246953887018, 5.39612581868729237315379234562, 6.158223864543346045003162522972, 7.1440596465996538372880172536, 8.50421688975096184540056256490, 9.173339289585727571235542244418, 10.04472972121759023492598158830, 10.820314328130689873971000159044, 11.70627099317166507740889780778, 12.33483532469541091859681798306, 13.212716877292964158580363895852, 13.76435312604597234126190964721, 14.58553978523501235202501508060, 15.32267727368671869580602085463, 16.01800399700840342206248008122, 17.28996355330168061018629938478, 17.77561931933993887064454351904, 18.70149870685269803744346327146, 19.49089323747921499448514387450, 20.30146524370781405149256014358, 21.06998831901854536246676966480

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