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Properties

Label 1-113-113.14-r0-0-0
Degree $1$
Conductor $113$
Sign $-0.426 + 0.904i$
Analytic cond. $0.524769$
Root an. cond. $0.524769$
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  + (0.623 + 0.781i)2-s + (0.433 + 0.900i)3-s + (−0.222 + 0.974i)4-s + (0.433 − 0.900i)5-s + (−0.433 + 0.900i)6-s + (−0.900 + 0.433i)7-s + (−0.900 + 0.433i)8-s + (−0.623 + 0.781i)9-s + (0.974 − 0.222i)10-s + (0.222 + 0.974i)11-s + (−0.974 + 0.222i)12-s + (0.900 − 0.433i)13-s + (−0.900 − 0.433i)14-s + 15-s + (−0.900 − 0.433i)16-s + (0.781 − 0.623i)17-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (0.433 + 0.900i)3-s + (−0.222 + 0.974i)4-s + (0.433 − 0.900i)5-s + (−0.433 + 0.900i)6-s + (−0.900 + 0.433i)7-s + (−0.900 + 0.433i)8-s + (−0.623 + 0.781i)9-s + (0.974 − 0.222i)10-s + (0.222 + 0.974i)11-s + (−0.974 + 0.222i)12-s + (0.900 − 0.433i)13-s + (−0.900 − 0.433i)14-s + 15-s + (−0.900 − 0.433i)16-s + (0.781 − 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $-0.426 + 0.904i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ -0.426 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8101157886 + 1.276959639i\)
\(L(\frac12)\) \(\approx\) \(0.8101157886 + 1.276959639i\)
\(L(1)\) \(\approx\) \(1.123696071 + 0.9531271116i\)
\(L(1)\) \(\approx\) \(1.123696071 + 0.9531271116i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad113 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 + (0.433 + 0.900i)T \)
5 \( 1 + (0.433 - 0.900i)T \)
7 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (0.781 - 0.623i)T \)
19 \( 1 + (-0.433 - 0.900i)T \)
23 \( 1 + (-0.433 + 0.900i)T \)
29 \( 1 + (0.781 - 0.623i)T \)
31 \( 1 + (0.900 - 0.433i)T \)
37 \( 1 + (0.974 - 0.222i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (-0.781 + 0.623i)T \)
47 \( 1 + (-0.974 - 0.222i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 + (-0.433 - 0.900i)T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 + (-0.974 + 0.222i)T \)
71 \( 1 - iT \)
73 \( 1 + iT \)
79 \( 1 + (0.974 + 0.222i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.781 - 0.623i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−29.360405157829563073412871670802, −28.58261035548461583832662532718, −26.90903050008990524925107312916, −25.91019763496120485261758720079, −24.93390323859018130777881351664, −23.56276931161458839542216225616, −23.02365099871812499267272900460, −21.825399947013906102697686739342, −20.83163694627757978895154128931, −19.51734060996232874643916926601, −18.93578855338510848986086537222, −18.14386239455940316080203580918, −16.45057852413289575633742667139, −14.73582878151826496929331036857, −13.96771858101706571926446357149, −13.23586079205767869347436298340, −12.11740217528544813848152580585, −10.85332047232672983744466447914, −9.84366023156139856200575793831, −8.38401947299444151594409415780, −6.44937637737470397871871389163, −6.16660535110317867077715378706, −3.68885800690569584888790928495, −2.92008632210569385144528559175, −1.38817470064942813335623970453, 2.72303508472264126792799780557, 4.08009300518996167561798167051, 5.161121125397502621402009422260, 6.22906092292713241056374838510, 7.95745236288993863810747382461, 9.092514993947361574590801418165, 9.84414477697225641841713774593, 11.84704721480515690866377026125, 13.03347568247676139441853177551, 13.83103231472277541333147108907, 15.25187312549143001284524757661, 15.86037172502624748085394422598, 16.77892803595150989551750225159, 17.850775829005210176781937829841, 19.694800996570982847892811306464, 20.74769907612496258198857864502, 21.50542832164694576110813360146, 22.55365974447916635827130968623, 23.41852079097124341717682341538, 25.0665819068618914418167320620, 25.38471055296460488673362096209, 26.24136937016266052543130968425, 27.71795734123194020912281300278, 28.35169999614840081921486879105, 29.90930748135142364398587056519

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