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Properties

Label 1-1157-1157.33-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.995 + 0.0916i$
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  + (−0.690 + 0.723i)2-s + (−0.672 − 0.739i)3-s + (−0.0475 − 0.998i)4-s + (0.654 − 0.755i)5-s + (0.999 + 0.0237i)6-s + (−0.436 + 0.899i)7-s + (0.755 + 0.654i)8-s + (−0.0950 + 0.995i)9-s + (0.0950 + 0.995i)10-s + (−0.945 − 0.327i)11-s + (−0.707 + 0.707i)12-s + (−0.349 − 0.936i)14-s + (−0.999 + 0.0237i)15-s + (−0.995 + 0.0950i)16-s + (−0.690 − 0.723i)17-s + (−0.654 − 0.755i)18-s + ⋯
L(s)  = 1  + (−0.690 + 0.723i)2-s + (−0.672 − 0.739i)3-s + (−0.0475 − 0.998i)4-s + (0.654 − 0.755i)5-s + (0.999 + 0.0237i)6-s + (−0.436 + 0.899i)7-s + (0.755 + 0.654i)8-s + (−0.0950 + 0.995i)9-s + (0.0950 + 0.995i)10-s + (−0.945 − 0.327i)11-s + (−0.707 + 0.707i)12-s + (−0.349 − 0.936i)14-s + (−0.999 + 0.0237i)15-s + (−0.995 + 0.0950i)16-s + (−0.690 − 0.723i)17-s + (−0.654 − 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.995 + 0.0916i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.995 + 0.0916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6395070596 + 0.02935593335i\)
\(L(\frac12)\) \(\approx\) \(0.6395070596 + 0.02935593335i\)
\(L(1)\) \(\approx\) \(0.5863193461 + 0.01930122518i\)
\(L(1)\) \(\approx\) \(0.5863193461 + 0.01930122518i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.690 + 0.723i)T \)
3 \( 1 + (-0.672 - 0.739i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (-0.436 + 0.899i)T \)
11 \( 1 + (-0.945 - 0.327i)T \)
17 \( 1 + (-0.690 - 0.723i)T \)
19 \( 1 + (-0.771 + 0.636i)T \)
23 \( 1 + (0.165 + 0.986i)T \)
29 \( 1 + (-0.560 - 0.828i)T \)
31 \( 1 + (0.349 + 0.936i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 + (0.304 + 0.952i)T \)
43 \( 1 + (0.560 - 0.828i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (-0.540 + 0.841i)T \)
59 \( 1 + (-0.739 - 0.672i)T \)
61 \( 1 + (0.992 - 0.118i)T \)
67 \( 1 + (0.998 + 0.0475i)T \)
71 \( 1 + (-0.981 + 0.189i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (-0.909 - 0.415i)T \)
83 \( 1 + (0.479 + 0.877i)T \)
97 \( 1 + (0.945 - 0.327i)T \)
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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

−21.12687978828159301404123778707, −20.65865570163272421318945361668, −19.791372735955238156299918600164, −18.862435264691722271100699595323, −18.09506242361530144866374903748, −17.41164343221880328237271473553, −16.943124693548433451602862685002, −16.061672767817427510579801392, −15.23337871534089612035117028876, −14.26441117981084112326569951553, −12.940502324323658004932757193907, −12.88600879243522857825854806117, −11.3568284631954707855464070196, −10.717867217799049908203315334488, −10.444717091107806567581950903475, −9.63788644309601552658478126421, −8.88055651683806276433917543153, −7.621348190302716501751462396267, −6.780204846390409465028413013528, −6.04229719409423737558937223710, −4.67445764146202455581919457551, −3.98051010833745579741993546506, −2.94544290222821616410227141212, −2.11251502302469507915062713967, −0.59733081647384800690265516588, 0.65038606676654847512111788603, 1.84811029419069962646048556548, 2.55525353294025595074118266448, 4.63403041564954948078925208421, 5.45932155212818977777417078492, 5.90854547569614553916290451671, 6.67321393868101181724761484116, 7.742216444044318818123018112580, 8.434226059863934171646506685107, 9.24880172719483547904961262156, 10.03939994906458754828656512639, 10.97164654584503290797180833231, 11.85552441721792078256793799729, 12.87356654255299059196555041092, 13.36023354863463280965602371863, 14.213261680365621321328835251140, 15.541876594986000660666423182495, 15.93468544804871415259169976453, 16.82638985408788889715780837562, 17.39387606954347145207501659979, 18.21249462813026183323886325497, 18.665201299649724847567111226465, 19.4230577991839890159406139991, 20.34552550614132015131532257313, 21.388007390252082305962523555343

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