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Properties

Label 1-1157-1157.1139-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.864 + 0.502i$
Analytic cond. $124.336$
Root an. cond. $124.336$
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  + (−0.989 − 0.142i)2-s + (−0.281 − 0.959i)3-s + (0.959 + 0.281i)4-s + (−0.909 − 0.415i)5-s + (0.142 + 0.989i)6-s + (0.415 − 0.909i)7-s + (−0.909 − 0.415i)8-s + (−0.841 + 0.540i)9-s + (0.841 + 0.540i)10-s + (0.909 − 0.415i)11-s − i·12-s + (−0.540 + 0.841i)14-s + (−0.142 + 0.989i)15-s + (0.841 + 0.540i)16-s + (−0.142 − 0.989i)17-s + (0.909 − 0.415i)18-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s + (−0.281 − 0.959i)3-s + (0.959 + 0.281i)4-s + (−0.909 − 0.415i)5-s + (0.142 + 0.989i)6-s + (0.415 − 0.909i)7-s + (−0.909 − 0.415i)8-s + (−0.841 + 0.540i)9-s + (0.841 + 0.540i)10-s + (0.909 − 0.415i)11-s − i·12-s + (−0.540 + 0.841i)14-s + (−0.142 + 0.989i)15-s + (0.841 + 0.540i)16-s + (−0.142 − 0.989i)17-s + (0.909 − 0.415i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.864 + 0.502i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.864 + 0.502i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2587346202 - 0.9593087294i\)
\(L(\frac12)\) \(\approx\) \(-0.2587346202 - 0.9593087294i\)
\(L(1)\) \(\approx\) \(0.4484740390 - 0.4322520338i\)
\(L(1)\) \(\approx\) \(0.4484740390 - 0.4322520338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.989 - 0.142i)T \)
3 \( 1 + (-0.281 - 0.959i)T \)
5 \( 1 + (-0.909 - 0.415i)T \)
7 \( 1 + (0.415 - 0.909i)T \)
11 \( 1 + (0.909 - 0.415i)T \)
17 \( 1 + (-0.142 - 0.989i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-0.540 - 0.841i)T \)
29 \( 1 + (-0.909 - 0.415i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 - T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + (0.281 - 0.959i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (-0.755 - 0.654i)T \)
67 \( 1 + (-0.281 - 0.959i)T \)
71 \( 1 + (0.909 - 0.415i)T \)
73 \( 1 + (0.540 - 0.841i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (0.909 + 0.415i)T \)
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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

−21.42196626046575805915954784111, −20.670246057616738349133524152630, −19.855917119747009180540354021058, −19.26264851262218646767286143032, −18.41015419578127481533843018876, −17.572511544408470789500894490794, −17.01998352384112992481472892816, −15.856336495717175341987303582284, −15.695817279642471137878276406923, −14.72608633457849088953971251090, −14.417028847127397523327593116232, −12.37547150611425632758232244106, −11.78074307630156466103542054012, −11.26895842999636838809216497833, −10.43309714599366395113348757012, −9.56672593438659503447632551468, −8.89470887283409610282986043084, −8.10831248617927455989550132305, −7.27345400459336677468961044918, −6.14977346596265211708700130857, −5.53189379005445954035498874837, −4.226014503500413775809785084836, −3.45026342631291095202147327650, −2.33707663942601955827580022045, −1.1025641832567660005586481968, 0.48250314811484979224262038513, 0.739050126484844173270659738300, 1.82566533381177760068164915096, 3.041946583170834203223123434156, 4.062720229306794619025008787715, 5.25622265736650870573857672655, 6.51472696838519205662016904242, 7.1801190506120946603688637861, 7.73962427242689751594816729475, 8.54369564015301690636928308735, 9.27356524969514859263956609953, 10.5281102233468470059403579935, 11.33079978290329998816877220997, 11.770185238634592825895375586988, 12.45442134766663611827339241142, 13.590508526146329447814154017612, 14.28538765823169215374381923095, 15.46757397318981469763943189650, 16.41403005099396918375349631787, 16.8022176528062799619239545121, 17.648306974500504778882541596953, 18.296755292579786744025785022826, 19.16592723981265002726590540299, 19.75781539384219197227353028334, 20.25180769679297857974591845027

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