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Properties

Label 1-1183-1183.684-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.750 - 0.661i$
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  + (−0.391 − 0.919i)2-s + (0.200 + 0.979i)3-s + (−0.692 + 0.721i)4-s + (−0.534 + 0.845i)5-s + (0.822 − 0.568i)6-s + (0.935 + 0.354i)8-s + (−0.919 + 0.391i)9-s + (0.987 + 0.160i)10-s + (−0.391 + 0.919i)11-s + (−0.845 − 0.534i)12-s + (−0.935 − 0.354i)15-s + (−0.0402 − 0.999i)16-s + (−0.632 − 0.774i)17-s + (0.721 + 0.692i)18-s + (−0.866 + 0.5i)19-s + (−0.239 − 0.970i)20-s + ⋯
L(s)  = 1  + (−0.391 − 0.919i)2-s + (0.200 + 0.979i)3-s + (−0.692 + 0.721i)4-s + (−0.534 + 0.845i)5-s + (0.822 − 0.568i)6-s + (0.935 + 0.354i)8-s + (−0.919 + 0.391i)9-s + (0.987 + 0.160i)10-s + (−0.391 + 0.919i)11-s + (−0.845 − 0.534i)12-s + (−0.935 − 0.354i)15-s + (−0.0402 − 0.999i)16-s + (−0.632 − 0.774i)17-s + (0.721 + 0.692i)18-s + (−0.866 + 0.5i)19-s + (−0.239 − 0.970i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.750 - 0.661i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (684, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.750 - 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03959843880 + 0.1048584539i\)
\(L(\frac12)\) \(\approx\) \(-0.03959843880 + 0.1048584539i\)
\(L(1)\) \(\approx\) \(0.5822575951 + 0.1275232424i\)
\(L(1)\) \(\approx\) \(0.5822575951 + 0.1275232424i\)

Euler product

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

−20.425458037547095985151074956995, −19.74971134876388234632026639470, −18.9760954361060548588797849079, −18.61807778058142330765424514132, −17.43282535170127115644413832259, −17.04430504521928912463382648670, −16.2082522966998907171717066634, −15.36880954677154615106375678165, −14.663424020680332370723631302249, −13.65457679195632165773123096522, −13.08508490709311829864734894969, −12.44935202104511411611215912341, −11.26985669194660331658319971326, −10.516785811076993532286677169868, −9.03972198806045197202129295381, −8.618925884045362897802691878569, −8.09364977197395597477995891604, −7.139645769082326571875607084507, −6.39345884077526014109435822189, −5.55276529881127799340785199857, −4.64916068164131797716561209997, −3.571214000869084818287903658077, −2.14119327201488322348096858571, −1.01926792440558509656612235012, −0.05801046302730259383734095872, 1.91958219305990582995472063040, 2.78716270184153686582264779560, 3.50665761690161744842285079012, 4.4043007025621605133369187767, 5.03628111738211371809255261859, 6.54749744331820521815900165897, 7.63425533070104081740484412247, 8.2763477345071041770862786868, 9.35391433688247205067875470148, 9.90418831258850530423870971744, 10.72213857576086231791721525565, 11.24092519331288915079830264434, 12.03041214489835736828697472182, 13.017014571583576682957311237686, 13.99829837445986703322570728647, 14.72556435418823004393102140796, 15.55532239479717687822539573170, 16.19579857009769090851114039317, 17.35466093399516865919625508006, 17.83857280657870635819006046462, 18.81350345697268678323407000659, 19.57945806171024969361957710346, 20.047318664107268510056608736296, 21.09647463189521787278423620124, 21.3409748573397218120419644569

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