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Properties

Label 1-1157-1157.54-r0-0-0
Degree $1$
Conductor $1157$
Sign $-0.330 - 0.943i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.330 - 0.943i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ -0.330 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3236206665 - 0.4560808081i\)
\(L(\frac12)\) \(\approx\) \(0.3236206665 - 0.4560808081i\)
\(L(1)\) \(\approx\) \(0.5001793250 - 0.1873644151i\)
\(L(1)\) \(\approx\) \(0.5001793250 - 0.1873644151i\)

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

−21.61630865172620078727500348799, −20.47739963948094452891662743640, −20.07321889775843699984121450769, −18.816998111382270032274367680156, −18.30201279142783323520001870832, −17.6426576835338184487521518162, −16.995038594890069452710214663771, −16.31889502074265565406996954745, −15.477639609277706714811038641051, −14.90319295630585370934606518517, −13.73233693600463193898771943886, −12.795105451893405617891430993, −11.82256736378812462073097092314, −10.94420722841582278930164616733, −10.35679180274098099038476040381, −9.72926044827153736017091120727, −9.19820634938447306367169536090, −7.641024498243703554430416367079, −6.8517672078746944427883428442, −6.50837487318237079090700863021, −5.558081544295756612056756028537, −4.52972243557780568712540541711, −3.17683853186041111205874294162, −2.17570587690064473334604602506, −0.946015657293671382713731055866, 0.457131780720762152400359825791, 1.422588541849444980054596952802, 2.43954499517472524381850490316, 3.52815176243889875190162253171, 5.063412912871618960738332492002, 5.80769377689245294487542941423, 6.49305498371599835262045833852, 7.33655086441807447224094851511, 8.537422263964340147858372973141, 9.11455098833257954201045402258, 9.93426063397655759337820171365, 10.73095088132393440694763012786, 11.57008255939089207172329802037, 12.24520711010416715134848203544, 13.08720547657188645510635386527, 13.56167520396008214281193011793, 15.29137159835712572160701372224, 16.13196378424268898596829240805, 16.43043099277825383730738320617, 17.25678340629087192098960476771, 18.04387384694026814373017446880, 18.4748089676890296374139051774, 19.47723317709690510994269089629, 19.98841920420181925643849681403, 21.17463713830412875437748428414

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