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Properties

Label 1-1157-1157.866-r0-0-0
Degree $1$
Conductor $1157$
Sign $-0.779 - 0.625i$
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  + (0.540 + 0.841i)2-s + (0.936 + 0.349i)3-s + (−0.415 + 0.909i)4-s + (0.142 + 0.989i)5-s + (0.212 + 0.977i)6-s + (−0.599 + 0.800i)7-s + (−0.989 + 0.142i)8-s + (0.755 + 0.654i)9-s + (−0.755 + 0.654i)10-s + (−0.989 − 0.142i)11-s + (−0.707 + 0.707i)12-s + (−0.997 − 0.0713i)14-s + (−0.212 + 0.977i)15-s + (−0.654 − 0.755i)16-s + (0.540 − 0.841i)17-s + (−0.142 + 0.989i)18-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)2-s + (0.936 + 0.349i)3-s + (−0.415 + 0.909i)4-s + (0.142 + 0.989i)5-s + (0.212 + 0.977i)6-s + (−0.599 + 0.800i)7-s + (−0.989 + 0.142i)8-s + (0.755 + 0.654i)9-s + (−0.755 + 0.654i)10-s + (−0.989 − 0.142i)11-s + (−0.707 + 0.707i)12-s + (−0.997 − 0.0713i)14-s + (−0.212 + 0.977i)15-s + (−0.654 − 0.755i)16-s + (0.540 − 0.841i)17-s + (−0.142 + 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.779 - 0.625i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (866, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ -0.779 - 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6519907409 + 1.854380485i\)
\(L(\frac12)\) \(\approx\) \(-0.6519907409 + 1.854380485i\)
\(L(1)\) \(\approx\) \(0.7972120280 + 1.287492039i\)
\(L(1)\) \(\approx\) \(0.7972120280 + 1.287492039i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.540 + 0.841i)T \)
3 \( 1 + (0.936 + 0.349i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (-0.599 + 0.800i)T \)
11 \( 1 + (-0.989 - 0.142i)T \)
17 \( 1 + (0.540 - 0.841i)T \)
19 \( 1 + (0.0713 + 0.997i)T \)
23 \( 1 + (-0.0713 - 0.997i)T \)
29 \( 1 + (-0.599 + 0.800i)T \)
31 \( 1 + (0.997 + 0.0713i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.936 + 0.349i)T \)
43 \( 1 + (0.599 + 0.800i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + (0.349 + 0.936i)T \)
61 \( 1 + (0.877 - 0.479i)T \)
67 \( 1 + (-0.909 + 0.415i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (0.755 - 0.654i)T \)
79 \( 1 + (-0.755 + 0.654i)T \)
83 \( 1 + (-0.212 - 0.977i)T \)
97 \( 1 + (0.989 - 0.142i)T \)
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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

−20.76012371967199594987184545796, −20.18481656092088587479380275981, −19.29625269070584284839811648527, −19.10306379788515060625388538070, −17.79219278701389253524256831911, −17.161355230280018299221354822540, −15.701365116075136618660418067915, −15.48714560316761006268231261062, −14.14472889181291092707468905279, −13.616243734811878007344864126092, −12.938648437058353908996162970700, −12.60805662661652516137727967857, −11.514957709131268971289258299010, −10.332344498398172286790731911, −9.79667550339482309598585327963, −9.02885510056266175463978366865, −8.12358847077080255748589209616, −7.23162874786076842182198071433, −6.06522897456880238891866495192, −5.11149613226884221914287884624, −4.14623234654059912578201568098, −3.482317859068747076156128326872, −2.47439823017510298123214003979, −1.56569525346971708285237493967, −0.554340981886660771445723077687, 2.21890850394501297753891299211, 3.02778556347161756715160278369, 3.4136775026910131288571725141, 4.71241122770824703321910826747, 5.572933936819038740074699648378, 6.45903700124005246464068748047, 7.327060949992746660962260969, 8.06651373253765822034738220410, 8.83112267030333629321202215497, 9.81493552635822357792076196385, 10.41303896975629172110800928581, 11.75737576659534573542018132254, 12.636205960760970082063558409894, 13.43708337210785081090018502486, 14.14842028354021735280785909132, 14.76506222955907919693358441608, 15.46485085684781579636591166464, 16.02366629307974728777683982769, 16.760595615660676755110184686593, 18.25137233121323553477265942272, 18.47870155721937105225162911527, 19.196667476092715975864009209379, 20.560864098902909779795434613857, 21.06286923139599637516510692403, 21.87899238213630117963408524766

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