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Properties

Label 1-113-113.18-r0-0-0
Degree $1$
Conductor $113$
Sign $0.0675 + 0.997i$
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  − 2-s + (0.707 + 0.707i)3-s + 4-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)6-s + 7-s − 8-s + i·9-s + (0.707 − 0.707i)10-s + i·11-s + (0.707 + 0.707i)12-s − i·13-s − 14-s − 15-s + 16-s + (−0.707 − 0.707i)17-s + ⋯
L(s)  = 1  − 2-s + (0.707 + 0.707i)3-s + 4-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)6-s + 7-s − 8-s + i·9-s + (0.707 − 0.707i)10-s + i·11-s + (0.707 + 0.707i)12-s − i·13-s − 14-s − 15-s + 16-s + (−0.707 − 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0675 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0675 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.0675 + 0.997i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.0675 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5821321574 + 0.5440642624i\)
\(L(\frac12)\) \(\approx\) \(0.5821321574 + 0.5440642624i\)
\(L(1)\) \(\approx\) \(0.7478231455 + 0.3442276636i\)
\(L(1)\) \(\approx\) \(0.7478231455 + 0.3442276636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad113 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + T \)
11 \( 1 + iT \)
13 \( 1 - iT \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + iT \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 - T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 - iT \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.707 - 0.707i)T \)
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

−28.95452726895139455514478965812, −28.08620886554356897784552613652, −26.887793208924007265497479845867, −26.36319805425888739160797240375, −24.931705806298057621612487645737, −24.15993199236396856907476053178, −23.7662838075766985643868191321, −21.32879200580606846397396127076, −20.66884819875455368761209979941, −19.42907343109466244423973283054, −19.053589581457604952276992858099, −17.73336864006571885306298632226, −16.79352713280780179866788462118, −15.52575860538255399326696540271, −14.51222795581732848133032504020, −13.06717687120117925488079746371, −11.79153877782736428059662722807, −11.00891273032275422656766469871, −8.99612432012380059524073231229, −8.53721574078655585110431518578, −7.584555399093425742731173578923, −6.31055972087903245493400452352, −4.222588741354566115737378646307, −2.415318346116893869277746251388, −1.05691033113520689085256734282, 2.07103049006808478734775091606, 3.3645643539379305510270030761, 4.94798871425586349555951424810, 7.071757610876868505799086352575, 7.93402963539756556649476469881, 8.89121486763041928090045926, 10.322980573145506106848210114272, 10.89874653779564414077138957786, 12.21486338542623596200279477433, 14.244442991914240899922148055005, 15.22591616519635359540660937931, 15.680372965464858386714529404633, 17.270309668664857368045510034451, 18.189606263374052639546413781934, 19.345273885538023359608299494152, 20.24946104956085444288933263693, 20.94338183847957349579746336714, 22.28846661362544073294412329066, 23.60230864445394658161161726593, 25.062833903536007274314974590797, 25.60745130894292302596263120033, 26.99447379323848941226700827201, 27.23505322625028462004733880547, 28.07255636072064670990126877899, 29.64489597205697243655915846504

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