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Properties

Label 1-1157-1157.77-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.918 - 0.394i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.918 - 0.394i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.918 - 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2424083956 - 1.179780519i\)
\(L(\frac12)\) \(\approx\) \(0.2424083956 - 1.179780519i\)
\(L(1)\) \(\approx\) \(0.7579331285 - 0.3610781161i\)
\(L(1)\) \(\approx\) \(0.7579331285 - 0.3610781161i\)

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

−21.31574307273195121645696346656, −20.30866663115831224483218038794, −19.77872472030470610090522810208, −19.30620883223563428595981858818, −18.42366650624924871586864075863, −17.61379222534204738774049270079, −16.68181986941553118472865469849, −16.09244077023064211278063526280, −15.36081856786616815123520372548, −14.5076508128626890467091717201, −13.95451147690249435259558532284, −12.87617152312644903355647022468, −11.41758966727319245162377085189, −11.08490870735589673984238134343, −10.1408466060960434728713196633, −9.39858437295391696564056267590, −9.11618175310990417041455991398, −7.6891524140968060577999711421, −7.18265140951741754484851427839, −6.489449570322238066551556498, −5.19472297135846933667928161059, −3.70239822252253714302961714956, −3.25446453699444853048645667790, −2.36129465886953895993586513482, −1.08171500476004182870479603330, 0.33081918955457673310669317839, 1.32937985726637811704143414535, 2.02826015332300376028762920887, 3.13281641934743911754460871616, 4.04980068872827780727561204510, 5.789396858059645181720987379226, 6.24642367263082754979793754360, 7.31478999881217408494398611159, 8.1223339645288752996387081355, 8.88940281649577125509992481030, 9.28370612764247728006866289367, 10.05810207073321750539587046196, 11.48956376540171195473825243547, 12.1710045296147305851925209225, 12.711498925823353574052211400706, 13.559865231664882066270394304245, 14.88219566734677661189066176621, 15.197449964075977402260672569316, 16.51299046371908373343177504257, 16.75818598664285460168277435921, 17.796688915712383026224062377423, 18.59366321325159026469889211874, 19.24348851118123331292300461441, 19.78234750742748458115387596556, 20.47310556623433559568839020988

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