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Properties

Label 1-1183-1183.324-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.673 + 0.739i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
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  + (0.632 − 0.774i)2-s + (0.428 + 0.903i)3-s + (−0.200 − 0.979i)4-s + (−0.799 + 0.600i)5-s + (0.970 + 0.239i)6-s + (−0.885 − 0.464i)8-s + (−0.632 + 0.774i)9-s + (−0.0402 + 0.999i)10-s + (0.632 + 0.774i)11-s + (0.799 − 0.600i)12-s + (−0.885 − 0.464i)15-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s + (0.200 + 0.979i)18-s + (0.5 + 0.866i)19-s + (0.748 + 0.663i)20-s + ⋯
L(s)  = 1  + (0.632 − 0.774i)2-s + (0.428 + 0.903i)3-s + (−0.200 − 0.979i)4-s + (−0.799 + 0.600i)5-s + (0.970 + 0.239i)6-s + (−0.885 − 0.464i)8-s + (−0.632 + 0.774i)9-s + (−0.0402 + 0.999i)10-s + (0.632 + 0.774i)11-s + (0.799 − 0.600i)12-s + (−0.885 − 0.464i)15-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s + (0.200 + 0.979i)18-s + (0.5 + 0.866i)19-s + (0.748 + 0.663i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.673 + 0.739i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.673 + 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3301192563 + 0.7472325916i\)
\(L(\frac12)\) \(\approx\) \(0.3301192563 + 0.7472325916i\)
\(L(1)\) \(\approx\) \(1.094619665 + 0.09032692192i\)
\(L(1)\) \(\approx\) \(1.094619665 + 0.09032692192i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.632 - 0.774i)T \)
3 \( 1 + (0.428 + 0.903i)T \)
5 \( 1 + (-0.799 + 0.600i)T \)
11 \( 1 + (0.632 + 0.774i)T \)
17 \( 1 + (-0.845 + 0.534i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.354 - 0.935i)T \)
31 \( 1 + (-0.692 + 0.721i)T \)
37 \( 1 + (-0.278 - 0.960i)T \)
41 \( 1 + (-0.568 + 0.822i)T \)
43 \( 1 + (-0.970 + 0.239i)T \)
47 \( 1 + (0.200 - 0.979i)T \)
53 \( 1 + (-0.845 + 0.534i)T \)
59 \( 1 + (0.919 + 0.391i)T \)
61 \( 1 + (-0.845 - 0.534i)T \)
67 \( 1 + (-0.948 + 0.316i)T \)
71 \( 1 + (-0.568 + 0.822i)T \)
73 \( 1 + (0.632 + 0.774i)T \)
79 \( 1 + (-0.200 + 0.979i)T \)
83 \( 1 + (-0.568 - 0.822i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.120 + 0.992i)T \)
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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

−20.79799563376380090608897921446, −20.163707851277525354276581579833, −19.506989793146477149728020987713, −18.5832578405455065279525279261, −17.76262129313064034365822878997, −16.99671294361737968570146927210, −16.20383298508170505019831837357, −15.45546941539680803947800845173, −14.74602097441594839661764486453, −13.72075549985765318094287230387, −13.404780268214386770548271806007, −12.50063709115412420688714033659, −11.69210772563701349410052743079, −11.29714724228580875882687671330, −9.21277031764324965285888671251, −8.8645812537206872458898710074, −7.970782167952803836665151435923, −7.28146126690036633721813136182, −6.5970362384823903578252741141, −5.61293748759796983681779758818, −4.70700543775280929141242309297, −3.63621916544735642463531537436, −3.060840388253350683670769533, −1.64956419343305474284189485425, −0.23520550423258113630677824892, 1.749815937222579065337858567822, 2.63755443158293123882974782249, 3.67281973662049963304825320869, 4.07404891992469660733225118160, 4.88596387717447272864275498891, 6.0180671346509482181904355832, 6.970936386779795996493853850308, 8.13739128668339751091387839397, 9.00932672937644922315982259112, 9.956381377444796175570354300717, 10.47827413465984521348663749717, 11.32730103087807967710773687069, 11.95886724822075036011969431505, 12.842413682375309250142772269691, 13.94596764478251989168439221677, 14.57645213600386142611618284703, 15.087732009662524253558680982534, 15.77344200614575145069838765458, 16.66346655294523651714998812205, 17.88424882354682783289114247271, 18.70395679006086728541470032815, 19.57820939042456587555337496752, 20.062709476891009514696045161940, 20.5889997545571386452305899551, 21.64639947562042645532033643340

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