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Properties

Label 2-1127-161.160-c1-0-26
Degree $2$
Conductor $1127$
Sign $0.998 - 0.0610i$
Analytic cond. $8.99914$
Root an. cond. $2.99985$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.14·2-s − 0.356i·3-s − 0.687·4-s − 3.71·5-s − 0.408i·6-s − 3.07·8-s + 2.87·9-s − 4.25·10-s + 4.66i·11-s + 0.245i·12-s − 4.84i·13-s + 1.32i·15-s − 2.15·16-s + 4.45·17-s + 3.29·18-s + 2.44·19-s + ⋯
L(s)  = 1  + 0.810·2-s − 0.205i·3-s − 0.343·4-s − 1.66·5-s − 0.166i·6-s − 1.08·8-s + 0.957·9-s − 1.34·10-s + 1.40i·11-s + 0.0707i·12-s − 1.34i·13-s + 0.342i·15-s − 0.538·16-s + 1.08·17-s + 0.775·18-s + 0.561·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.998 - 0.0610i$
Analytic conductor: \(8.99914\)
Root analytic conductor: \(2.99985\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (1126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :1/2),\ 0.998 - 0.0610i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.561034238\)
\(L(\frac12)\) \(\approx\) \(1.561034238\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 + (-4.24 + 2.22i)T \)
good2 \( 1 - 1.14T + 2T^{2} \)
3 \( 1 + 0.356iT - 3T^{2} \)
5 \( 1 + 3.71T + 5T^{2} \)
11 \( 1 - 4.66iT - 11T^{2} \)
13 \( 1 + 4.84iT - 13T^{2} \)
17 \( 1 - 4.45T + 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
29 \( 1 - 4.72T + 29T^{2} \)
31 \( 1 - 7.38iT - 31T^{2} \)
37 \( 1 - 9.44iT - 37T^{2} \)
41 \( 1 - 1.19iT - 41T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 - 0.582iT - 47T^{2} \)
53 \( 1 - 5.02iT - 53T^{2} \)
59 \( 1 + 3.07iT - 59T^{2} \)
61 \( 1 + 3.98T + 61T^{2} \)
67 \( 1 - 1.21iT - 67T^{2} \)
71 \( 1 - 3.80T + 71T^{2} \)
73 \( 1 + 5.46iT - 73T^{2} \)
79 \( 1 + 6.81iT - 79T^{2} \)
83 \( 1 - 4.38T + 83T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.01445582335637621473304809509, −8.879079881077794633170746917719, −7.949452140349229744914640808905, −7.41873288386282680589003575993, −6.58534383665263458277705158272, −5.05366613571504617574221488281, −4.74175722871846817105613702172, −3.69048414746494935624593110354, −3.02682714622161532732621530876, −0.919076820947213606783839813922, 0.818287458822628619111552207969, 3.11276779280786072737404836683, 3.80732745085933121426551706171, 4.37199195998075767901402140935, 5.28567246401638068526307902475, 6.37482103767411646258485036747, 7.38202341330432467674394392877, 8.091618295814695094047753101073, 9.001607938001094738040625767952, 9.664534091538897121067166840031

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