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Properties

Label 1-1157-1157.330-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: $\chi_{1157} (330, \cdot )$
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

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

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