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Properties

Label 1-1183-1183.734-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.263 + 0.964i$
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  + (−0.0804 + 0.996i)2-s + (0.845 + 0.534i)3-s + (−0.987 − 0.160i)4-s + (0.663 − 0.748i)5-s + (−0.600 + 0.799i)6-s + (0.239 − 0.970i)8-s + (0.428 + 0.903i)9-s + (0.692 + 0.721i)10-s + (0.903 + 0.428i)11-s + (−0.748 − 0.663i)12-s + (0.960 − 0.278i)15-s + (0.948 + 0.316i)16-s + (0.692 − 0.721i)17-s + (−0.935 + 0.354i)18-s + (−0.866 − 0.5i)19-s + (−0.774 + 0.632i)20-s + ⋯
L(s)  = 1  + (−0.0804 + 0.996i)2-s + (0.845 + 0.534i)3-s + (−0.987 − 0.160i)4-s + (0.663 − 0.748i)5-s + (−0.600 + 0.799i)6-s + (0.239 − 0.970i)8-s + (0.428 + 0.903i)9-s + (0.692 + 0.721i)10-s + (0.903 + 0.428i)11-s + (−0.748 − 0.663i)12-s + (0.960 − 0.278i)15-s + (0.948 + 0.316i)16-s + (0.692 − 0.721i)17-s + (−0.935 + 0.354i)18-s + (−0.866 − 0.5i)19-s + (−0.774 + 0.632i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.263 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.775435325 + 1.355796517i\)
\(L(\frac12)\) \(\approx\) \(1.775435325 + 1.355796517i\)
\(L(1)\) \(\approx\) \(1.293170279 + 0.7246790593i\)
\(L(1)\) \(\approx\) \(1.293170279 + 0.7246790593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.0804 + 0.996i)T \)
3 \( 1 + (0.845 + 0.534i)T \)
5 \( 1 + (0.663 - 0.748i)T \)
11 \( 1 + (0.903 + 0.428i)T \)
17 \( 1 + (0.692 - 0.721i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.996 - 0.0804i)T \)
31 \( 1 + (0.992 + 0.120i)T \)
37 \( 1 + (-0.391 - 0.919i)T \)
41 \( 1 + (0.534 - 0.845i)T \)
43 \( 1 + (0.919 + 0.391i)T \)
47 \( 1 + (0.935 + 0.354i)T \)
53 \( 1 + (-0.970 - 0.239i)T \)
59 \( 1 + (-0.316 - 0.948i)T \)
61 \( 1 + (-0.278 + 0.960i)T \)
67 \( 1 + (-0.160 - 0.987i)T \)
71 \( 1 + (0.999 + 0.0402i)T \)
73 \( 1 + (0.822 - 0.568i)T \)
79 \( 1 + (-0.354 + 0.935i)T \)
83 \( 1 + (-0.464 + 0.885i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
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

−21.04148787601579290094989033203, −20.30259749408891481833499845069, −19.35268978561750481185703561666, −18.86133384334011271334919088125, −18.45963180944099781949058925022, −17.28146104204583687530420499764, −16.95588743157969108119437194144, −15.221359664420748596228268862446, −14.43001527715412182345046375231, −14.13395209559139097715945432012, −13.17897232420407624223674966537, −12.57088522738297529049959873238, −11.695187969583538088625094104629, −10.72533346488510084645031662965, −10.04111753449479385913373350830, −9.21528832962585359042538334689, −8.53717983180363335430266853231, −7.66807829031154232867075390228, −6.53594328930956884120736253120, −5.84497615752974303582944748083, −4.33312063588433675333017677782, −3.51319611013665117216610447326, −2.77417014935389566714644855869, −1.89995115025475160022383205620, −1.11937174474741224109792600839, 1.08201529241101362286133016559, 2.21899336760534718224952565549, 3.5849946590932582598047384133, 4.42151817185822559454287928637, 5.1194512652305169758811505521, 6.00354086004358869876637539057, 7.09009932605705832287295382437, 7.83822362323545132580582303850, 8.837557013303456785350882551524, 9.31407023561626292357179035930, 9.79722823261603835170646983543, 10.90885030247603424376937394939, 12.37788524173585873172889200111, 13.04254301174207110239319386019, 14.02540687297046480091950238111, 14.24538993554914006152316291152, 15.34028018343335920902613755863, 15.82024158728041707844827697429, 16.894374819271485502651824947769, 17.138214831335373227104024304904, 18.13709078934870610868139082872, 19.21283027516239268774928135968, 19.68464047070010269100215246195, 20.847384319161731598743303856231, 21.24273603361857439337672466928

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