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Properties

Label 1-1183-1183.227-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.787 + 0.615i$
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  + (−0.822 − 0.568i)2-s + (0.0402 − 0.999i)3-s + (0.354 + 0.935i)4-s + (0.316 + 0.948i)5-s + (−0.600 + 0.799i)6-s + (0.239 − 0.970i)8-s + (−0.996 − 0.0804i)9-s + (0.278 − 0.960i)10-s + (−0.0804 − 0.996i)11-s + (0.948 − 0.316i)12-s + (0.960 − 0.278i)15-s + (−0.748 + 0.663i)16-s + (−0.970 − 0.239i)17-s + (0.774 + 0.632i)18-s + (0.866 − 0.5i)19-s + (−0.774 + 0.632i)20-s + ⋯
L(s)  = 1  + (−0.822 − 0.568i)2-s + (0.0402 − 0.999i)3-s + (0.354 + 0.935i)4-s + (0.316 + 0.948i)5-s + (−0.600 + 0.799i)6-s + (0.239 − 0.970i)8-s + (−0.996 − 0.0804i)9-s + (0.278 − 0.960i)10-s + (−0.0804 − 0.996i)11-s + (0.948 − 0.316i)12-s + (0.960 − 0.278i)15-s + (−0.748 + 0.663i)16-s + (−0.970 − 0.239i)17-s + (0.774 + 0.632i)18-s + (0.866 − 0.5i)19-s + (−0.774 + 0.632i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.787 + 0.615i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.787 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08165013380 - 0.2371022590i\)
\(L(\frac12)\) \(\approx\) \(-0.08165013380 - 0.2371022590i\)
\(L(1)\) \(\approx\) \(0.5256423814 - 0.2857685759i\)
\(L(1)\) \(\approx\) \(0.5256423814 - 0.2857685759i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.822 - 0.568i)T \)
3 \( 1 + (0.0402 - 0.999i)T \)
5 \( 1 + (0.316 + 0.948i)T \)
11 \( 1 + (-0.0804 - 0.996i)T \)
17 \( 1 + (-0.970 - 0.239i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.996 - 0.0804i)T \)
31 \( 1 + (-0.391 - 0.919i)T \)
37 \( 1 + (0.992 + 0.120i)T \)
41 \( 1 + (0.534 - 0.845i)T \)
43 \( 1 + (0.919 + 0.391i)T \)
47 \( 1 + (-0.774 + 0.632i)T \)
53 \( 1 + (0.278 + 0.960i)T \)
59 \( 1 + (-0.663 + 0.748i)T \)
61 \( 1 + (-0.692 - 0.721i)T \)
67 \( 1 + (-0.774 + 0.632i)T \)
71 \( 1 + (0.999 + 0.0402i)T \)
73 \( 1 + (-0.903 - 0.428i)T \)
79 \( 1 + (-0.632 - 0.774i)T \)
83 \( 1 + (-0.464 + 0.885i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.979 - 0.200i)T \)
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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

−21.543962514191361113529595823012, −20.68236421070871145939422304114, −20.019706478126283919930278649397, −19.77708314642228701666974512943, −18.250731888238274827401871406691, −17.73270101040689745238262389056, −16.97837386444790364196721337930, −16.24001344905298025825063553164, −15.80302055304851073246426362480, −14.922099594635632866859454187611, −14.24768159090015806345810914909, −13.25350199494359364844388049743, −12.145726048165914724816217842882, −11.25805097172480016802418012554, −10.330749692421737450120454054587, −9.62222120210744784977991488863, −9.18315400699289891122920488815, −8.30222068536998009968671754217, −7.54498107841525887114488934981, −6.31242524940938353951809308112, −5.49626772265263764811216072598, −4.79656464278676903888308771726, −3.96243319597736914361309207776, −2.398431876330670349899464154971, −1.48335433651440260744317960112, 0.129303485349579792567945037074, 1.41117374573362106650860693995, 2.43394325582259444378589726772, 2.93239259767639777042861272991, 4.008245354458420326791276556121, 5.79011580624971215374267682105, 6.37705271481465688971444307435, 7.41797729609481072841187553349, 7.78153290823249206158685082712, 8.93900855638672639565037172657, 9.53407279784703114099405278882, 10.76168036652143941840337675535, 11.24664779235437358526908962167, 11.85394328474775366488568986398, 13.00592374996071822802570112380, 13.55538643822269667374380498767, 14.295368392800754148792469481928, 15.44748830076298271283184732565, 16.35991715029073881151766628226, 17.27028500747785548164655415032, 17.98003430347118652834453799629, 18.43833441065502199102508603637, 19.05754573445383289362292048962, 19.819976605191053985406652632802, 20.45126778264931197755882244014

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