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Properties

Label 1-1183-1183.872-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.533 + 0.845i$
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  + (−0.200 − 0.979i)2-s + (−0.632 + 0.774i)3-s + (−0.919 + 0.391i)4-s + (0.278 − 0.960i)5-s + (0.885 + 0.464i)6-s + (0.568 + 0.822i)8-s + (−0.200 − 0.979i)9-s + (−0.996 − 0.0804i)10-s + (−0.200 + 0.979i)11-s + (0.278 − 0.960i)12-s + (0.568 + 0.822i)15-s + (0.692 − 0.721i)16-s + (0.428 − 0.903i)17-s + (−0.919 + 0.391i)18-s + (−0.5 + 0.866i)19-s + (0.120 + 0.992i)20-s + ⋯
L(s)  = 1  + (−0.200 − 0.979i)2-s + (−0.632 + 0.774i)3-s + (−0.919 + 0.391i)4-s + (0.278 − 0.960i)5-s + (0.885 + 0.464i)6-s + (0.568 + 0.822i)8-s + (−0.200 − 0.979i)9-s + (−0.996 − 0.0804i)10-s + (−0.200 + 0.979i)11-s + (0.278 − 0.960i)12-s + (0.568 + 0.822i)15-s + (0.692 − 0.721i)16-s + (0.428 − 0.903i)17-s + (−0.919 + 0.391i)18-s + (−0.5 + 0.866i)19-s + (0.120 + 0.992i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.533 + 0.845i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (872, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.533 + 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03558809049 + 0.06450344084i\)
\(L(\frac12)\) \(\approx\) \(0.03558809049 + 0.06450344084i\)
\(L(1)\) \(\approx\) \(0.5673770492 - 0.1773145399i\)
\(L(1)\) \(\approx\) \(0.5673770492 - 0.1773145399i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.200 - 0.979i)T \)
3 \( 1 + (-0.632 + 0.774i)T \)
5 \( 1 + (0.278 - 0.960i)T \)
11 \( 1 + (-0.200 + 0.979i)T \)
17 \( 1 + (0.428 - 0.903i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.748 + 0.663i)T \)
31 \( 1 + (-0.0402 - 0.999i)T \)
37 \( 1 + (-0.845 + 0.534i)T \)
41 \( 1 + (-0.354 - 0.935i)T \)
43 \( 1 + (0.885 - 0.464i)T \)
47 \( 1 + (-0.919 - 0.391i)T \)
53 \( 1 + (0.428 - 0.903i)T \)
59 \( 1 + (0.692 + 0.721i)T \)
61 \( 1 + (0.428 + 0.903i)T \)
67 \( 1 + (0.799 - 0.600i)T \)
71 \( 1 + (-0.354 - 0.935i)T \)
73 \( 1 + (-0.200 + 0.979i)T \)
79 \( 1 + (-0.919 - 0.391i)T \)
83 \( 1 + (-0.354 + 0.935i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.970 - 0.239i)T \)
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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

−21.420911019077879469083082555749, −19.72198033127918238396228166249, −18.97146979314310886828356416826, −18.660811528340989762594257370852, −17.70552500472494611314497715936, −17.302621090835189487200450951981, −16.40596411820755625725169565664, −15.691404231240011549969418932, −14.62606350102481317460898007387, −14.11550751204916234267222498642, −13.267052837694351258157124212944, −12.64301178652903527885295569092, −11.35739804421936896132974659625, −10.70428320213719629302927433141, −9.98914338228099585802925276571, −8.71099270953233859964757661239, −8.01926953547577319373886448583, −7.17924101211361591193870166805, −6.3927459413864192604933758018, −5.96458269122575343771668663354, −5.082200349273735104076007197188, −3.85420642598469890413742309820, −2.62739529441153837500626601306, −1.41458835558560399537408871643, −0.037545141076371622193470245949, 1.29144546396423311592373975581, 2.2352712001912682074406338368, 3.625449576968061176164502872461, 4.245146773270352283383083618844, 5.20275955204313323021770993574, 5.60952066815252899107463397789, 7.175256754091159058684075612227, 8.2660765207156695034398594445, 9.14746592483482030299734956725, 9.82357988271660639647930015651, 10.24321000372855596150622208253, 11.36495235458828601687458460418, 12.04641491469454659541153619360, 12.5819473978334830645377408544, 13.46407523540987072051822953917, 14.42515970929609016439517919072, 15.398124220862191967086606507133, 16.37595473470146945050022308897, 16.927509293771175534923591670787, 17.66944830439434074306225535559, 18.26535117943250028527973829967, 19.32183724344853861121964368200, 20.34978373269217216526418679425, 20.70571570928926607888636253129, 21.20277941367315260356272951140

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