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Properties

Label 1-1157-1157.384-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.183 + 0.983i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (−0.971 + 0.235i)2-s + (0.952 − 0.304i)3-s + (0.888 − 0.458i)4-s + (0.654 + 0.755i)5-s + (−0.853 + 0.520i)6-s + (0.828 + 0.560i)7-s + (−0.755 + 0.654i)8-s + (0.814 − 0.580i)9-s + (−0.814 − 0.580i)10-s + (−0.189 + 0.981i)11-s + (0.707 − 0.707i)12-s + (−0.936 − 0.349i)14-s + (0.853 + 0.520i)15-s + (0.580 − 0.814i)16-s + (−0.971 − 0.235i)17-s + (−0.654 + 0.755i)18-s + ⋯
L(s)  = 1  + (−0.971 + 0.235i)2-s + (0.952 − 0.304i)3-s + (0.888 − 0.458i)4-s + (0.654 + 0.755i)5-s + (−0.853 + 0.520i)6-s + (0.828 + 0.560i)7-s + (−0.755 + 0.654i)8-s + (0.814 − 0.580i)9-s + (−0.814 − 0.580i)10-s + (−0.189 + 0.981i)11-s + (0.707 − 0.707i)12-s + (−0.936 − 0.349i)14-s + (0.853 + 0.520i)15-s + (0.580 − 0.814i)16-s + (−0.971 − 0.235i)17-s + (−0.654 + 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.183 + 0.983i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (384, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.183 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.230408839 + 1.021911631i\)
\(L(\frac12)\) \(\approx\) \(1.230408839 + 1.021911631i\)
\(L(1)\) \(\approx\) \(1.076775080 + 0.3399890583i\)
\(L(1)\) \(\approx\) \(1.076775080 + 0.3399890583i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.971 + 0.235i)T \)
3 \( 1 + (0.952 - 0.304i)T \)
5 \( 1 + (0.654 + 0.755i)T \)
7 \( 1 + (0.828 + 0.560i)T \)
11 \( 1 + (-0.189 + 0.981i)T \)
17 \( 1 + (-0.971 - 0.235i)T \)
19 \( 1 + (-0.986 - 0.165i)T \)
23 \( 1 + (-0.636 + 0.771i)T \)
29 \( 1 + (-0.899 + 0.436i)T \)
31 \( 1 + (0.936 + 0.349i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (0.739 - 0.672i)T \)
43 \( 1 + (0.899 + 0.436i)T \)
47 \( 1 + (0.841 + 0.540i)T \)
53 \( 1 + (0.540 + 0.841i)T \)
59 \( 1 + (-0.304 + 0.952i)T \)
61 \( 1 + (0.919 - 0.393i)T \)
67 \( 1 + (0.458 - 0.888i)T \)
71 \( 1 + (0.327 + 0.945i)T \)
73 \( 1 + (-0.909 + 0.415i)T \)
79 \( 1 + (0.909 - 0.415i)T \)
83 \( 1 + (-0.877 - 0.479i)T \)
97 \( 1 + (0.189 + 0.981i)T \)
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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

−20.8215922460111585026714279285, −20.48086100594500559121365294502, −19.585530480110632639402218272254, −18.9386808224743652744433784282, −18.02311716033385731586056786038, −17.23997600552615700846180644073, −16.62521153175658559336364422366, −15.849986605986955197735703912, −14.99295271224387522722984697297, −14.03676119017134490396070862513, −13.34917730610664321955744638236, −12.56856333022657393663400867910, −11.36102688564946373010509435644, −10.58851347568073662535380772298, −9.99100958904618609488482360832, −8.97470689860186038147890687149, −8.44770323242902475864312560621, −7.98631937259270788794350419703, −6.83138007942781233423610119452, −5.810146811448336422953251346783, −4.50798066150927418895564568336, −3.796009300629461435259021230180, −2.40317899461364935584732906145, −1.92390885883364429400427937942, −0.752974529516943865693599120224, 1.51723501964911712395678719302, 2.193752481134545103564745649573, 2.67230569068474251485477265049, 4.168736862486230180384845099688, 5.45046154918789919793265651320, 6.43985097024717746792982015477, 7.22419989990199065289180025132, 7.80728263095597796346042438901, 8.86800471645483069180282353451, 9.27472052694570601773003469348, 10.25408144729427728389126199528, 10.91408028408875741389369871447, 11.93686305608466353324267825464, 12.86607770007837341278268574143, 13.99608964689148128636541911690, 14.53720808165876152965678968420, 15.37147962772706372086709498528, 15.648055853220718851493198712836, 17.35103623724258586863291171287, 17.66013130905568045315996881088, 18.330901143258289928456647274215, 19.00785751094499156679763055118, 19.74568391190779585877259909086, 20.62271987797656168815778678916, 21.13733155566089555434449382502

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