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Properties

Label 1-1157-1157.675-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.994 + 0.100i$
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  − 2-s + (−0.707 − 0.707i)3-s + 4-s + i·5-s + (0.707 + 0.707i)6-s + (0.707 + 0.707i)7-s − 8-s + i·9-s − i·10-s + 11-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)14-s + (0.707 − 0.707i)15-s + 16-s − i·17-s − i·18-s + ⋯
L(s)  = 1  − 2-s + (−0.707 − 0.707i)3-s + 4-s + i·5-s + (0.707 + 0.707i)6-s + (0.707 + 0.707i)7-s − 8-s + i·9-s − i·10-s + 11-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)14-s + (0.707 − 0.707i)15-s + 16-s − i·17-s − i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (675, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.994 + 0.100i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01250271009 + 0.2478568358i\)
\(L(\frac12)\) \(\approx\) \(0.01250271009 + 0.2478568358i\)
\(L(1)\) \(\approx\) \(0.5622349348 + 0.05189172092i\)
\(L(1)\) \(\approx\) \(0.5622349348 + 0.05189172092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + iT \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + T \)
17 \( 1 - iT \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + iT \)
53 \( 1 - iT \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 - T \)
71 \( 1 + iT \)
73 \( 1 + T \)
79 \( 1 + iT \)
83 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 - T \)
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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

−20.60517508230386097481083292275, −19.901638457278533158797946556459, −19.42485190872758773652182051319, −18.00928553858390418087302708909, −17.297898869966958880796211774437, −17.146215641261704678245357763167, −16.32139485096787977351307617567, −15.56757876264866619898204193975, −14.80452703173539929887548077581, −13.76762068377419383727319924954, −12.387294708085437450187985324473, −11.962466897821503206877593915567, −11.012485807309137598111269160272, −10.4557511649122029654948346492, −9.57159892483424536347175231315, −8.793066216243663255908814337302, −8.205426241387687409460676122799, −7.00580335011702837768612607239, −6.237548143537354476139655620877, −5.24112353630188571525266007427, −4.326921782248230626004479597, −3.55890730892933107201995235990, −1.76310150389506701403069861921, −1.08994457893913973627115358139, −0.088470492203278573993212647387, 1.16694138167566379145777942969, 2.10774743994939443810775595351, 2.76733149071925734062832696643, 4.30691993353464459699381228647, 5.690886259475388000036782853054, 6.34458350322312393834530931238, 6.95149732550308750851922328864, 7.90088071656521156067083544468, 8.49511172442248039111574233201, 9.68598234452367133213519418683, 10.45192485456525944878065034490, 11.31636444853169976254809832682, 11.8074125016009192129248093169, 12.35033359986031898565855748087, 13.85349404447327337974418662497, 14.51515583878176045907371185831, 15.38499479659994531997550011320, 16.23837541254205957245587621129, 17.18142220305319983695843076907, 17.6923080599576544603175220343, 18.37905728562898864324075900003, 18.951070750870058439763316592513, 19.450998079321124603381931864279, 20.62441113507702189198734549890, 21.41387186706618712174424541406

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