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Properties

Label 1-1157-1157.343-r0-0-0
Degree $1$
Conductor $1157$
Sign $-0.153 - 0.988i$
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  + (0.909 − 0.415i)2-s + (0.0713 − 0.997i)3-s + (0.654 − 0.755i)4-s + (0.959 + 0.281i)5-s + (−0.349 − 0.936i)6-s + (0.877 − 0.479i)7-s + (0.281 − 0.959i)8-s + (−0.989 − 0.142i)9-s + (0.989 − 0.142i)10-s + (0.281 + 0.959i)11-s + (−0.707 − 0.707i)12-s + (0.599 − 0.800i)14-s + (0.349 − 0.936i)15-s + (−0.142 − 0.989i)16-s + (0.909 + 0.415i)17-s + (−0.959 + 0.281i)18-s + ⋯
L(s)  = 1  + (0.909 − 0.415i)2-s + (0.0713 − 0.997i)3-s + (0.654 − 0.755i)4-s + (0.959 + 0.281i)5-s + (−0.349 − 0.936i)6-s + (0.877 − 0.479i)7-s + (0.281 − 0.959i)8-s + (−0.989 − 0.142i)9-s + (0.989 − 0.142i)10-s + (0.281 + 0.959i)11-s + (−0.707 − 0.707i)12-s + (0.599 − 0.800i)14-s + (0.349 − 0.936i)15-s + (−0.142 − 0.989i)16-s + (0.909 + 0.415i)17-s + (−0.959 + 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.153 - 0.988i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ -0.153 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.421796395 - 2.827785082i\)
\(L(\frac12)\) \(\approx\) \(2.421796395 - 2.827785082i\)
\(L(1)\) \(\approx\) \(1.938038718 - 1.279680544i\)
\(L(1)\) \(\approx\) \(1.938038718 - 1.279680544i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.909 - 0.415i)T \)
3 \( 1 + (0.0713 - 0.997i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (0.877 - 0.479i)T \)
11 \( 1 + (0.281 + 0.959i)T \)
17 \( 1 + (0.909 + 0.415i)T \)
19 \( 1 + (-0.800 + 0.599i)T \)
23 \( 1 + (0.800 - 0.599i)T \)
29 \( 1 + (0.877 - 0.479i)T \)
31 \( 1 + (-0.599 + 0.800i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.0713 - 0.997i)T \)
43 \( 1 + (-0.877 - 0.479i)T \)
47 \( 1 + (-0.654 + 0.755i)T \)
53 \( 1 + (-0.755 + 0.654i)T \)
59 \( 1 + (0.997 - 0.0713i)T \)
61 \( 1 + (-0.977 - 0.212i)T \)
67 \( 1 + (-0.755 + 0.654i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (-0.989 + 0.142i)T \)
79 \( 1 + (0.989 - 0.142i)T \)
83 \( 1 + (0.349 + 0.936i)T \)
97 \( 1 + (-0.281 + 0.959i)T \)
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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

−21.593658593588458629119716832210, −21.06852553193423937185554870962, −20.435392286856465562621819365168, −19.40120093322114021573445196839, −18.14570601922330085607882619565, −17.26540638783850791027265923513, −16.72920745721153893022727065862, −16.076145866031883613763709570564, −15.00640464623845972582546131386, −14.64062207849992746473378291920, −13.81353836041291256918164454950, −13.20677864440468668317167243787, −12.01207767120359929444150394854, −11.32631006488394396379211334671, −10.617199231528930833431685259276, −9.50135099697073374896872900821, −8.65899759513739909932621218810, −8.09130774805232450671162462617, −6.662869954124206913659956802412, −5.816644853148532544832510512065, −5.135178413477520333110458721244, −4.661436651465193621202683163267, −3.396368505694927336452562620315, −2.7144014589727206502288185091, −1.543991199148913115255671547013, 1.263554984453559635519738623450, 1.77521601251058747592246689097, 2.59622482277474140457851873112, 3.71582894945128653573439123304, 4.86228646921862529357646042535, 5.58762839699657038098067463268, 6.553679360159252542627254122, 7.07297216500589841577651611502, 8.07400805208234610164814437785, 9.22604618800826739870155165478, 10.40136550890857649299902466774, 10.76154482895471565072469046956, 11.97130264388034215785264725864, 12.49242634982222745188170060419, 13.22549874136182274935622950278, 14.162429549926207475914695849, 14.42267059655797673135729006077, 15.11733425672517967283647247905, 16.65313301300512880295197564269, 17.33632521345689310479457391538, 18.012533359928924367419255753324, 18.89750784006985046422807172929, 19.5436529984323994714765771101, 20.624856831168027771676093477121, 20.884204166779617116813446452961

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