Properties

Degree 1
Conductor 113
Sign $-0.751 + 0.659i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.623 − 0.781i)2-s + (−0.330 + 0.943i)3-s + (−0.222 + 0.974i)4-s + (0.330 + 0.943i)5-s + (0.943 − 0.330i)6-s + (−0.900 + 0.433i)7-s + (0.900 − 0.433i)8-s + (−0.781 − 0.623i)9-s + (0.532 − 0.846i)10-s + (−0.974 + 0.222i)11-s + (−0.846 − 0.532i)12-s + (−0.433 − 0.900i)13-s + (0.900 + 0.433i)14-s − 15-s + (−0.900 − 0.433i)16-s + (−0.993 − 0.111i)17-s + ⋯
L(s,χ)  = 1  + (−0.623 − 0.781i)2-s + (−0.330 + 0.943i)3-s + (−0.222 + 0.974i)4-s + (0.330 + 0.943i)5-s + (0.943 − 0.330i)6-s + (−0.900 + 0.433i)7-s + (0.900 − 0.433i)8-s + (−0.781 − 0.623i)9-s + (0.532 − 0.846i)10-s + (−0.974 + 0.222i)11-s + (−0.846 − 0.532i)12-s + (−0.433 − 0.900i)13-s + (0.900 + 0.433i)14-s − 15-s + (−0.900 − 0.433i)16-s + (−0.993 − 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.751 + 0.659i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.751 + 0.659i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(113\)
\( \varepsilon \)  =  $-0.751 + 0.659i$
motivic weight  =  \(0\)
character  :  $\chi_{113} (26, \cdot )$
Sato-Tate  :  $\mu(56)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 113,\ (0:\ ),\ -0.751 + 0.659i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1198745445 + 0.3185980874i$
$L(\frac12,\chi)$  $\approx$  $0.1198745445 + 0.3185980874i$
$L(\chi,1)$  $\approx$  0.4847767321 + 0.1648484067i
$L(1,\chi)$  $\approx$  0.4847767321 + 0.1648484067i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−28.74594601872304859118155105727, −28.36167372723832943800080489716, −26.6257544289741631748333777545, −25.91314947940465500744465284712, −24.72604124334560739653179448349, −24.101536704268350622240958029051, −23.32032862129071582806751171902, −22.08305547912523246907978511510, −20.245869720307041829021585076846, −19.47444479409129313443377810853, −18.38715093631060841298938456406, −17.43200737650659766481145015117, −16.528455984622549008326932622, −15.779757303481586226767829579731, −13.814322403724162388714360721129, −13.35411510613456384945531168420, −11.988487414427533042337894661620, −10.40766214349040327944744527759, −9.223186318716256146314607383396, −8.09789032996142407789581716269, −6.95110698816818717295797251363, −5.97059073589228201705005530738, −4.7670817855748935920689119254, −2.0903882185246314129381545262, −0.39294154494863636215823262169, 2.622005134230389225016760273928, 3.38635285140839936519379948784, 5.155466776414635689306897771309, 6.677354614004525887281880054466, 8.275128193666022343110257748989, 9.87368093977100414984960760441, 10.06996982881667267318873551669, 11.2738519588939764789834051756, 12.44130321182344325502823938784, 13.76438872653153011000511303911, 15.41192946208901844987405356454, 16.08808257326222114978565953479, 17.63474156130246699326468068372, 18.15468232810773712178107087162, 19.50746421102256018772822846578, 20.468991283597460729247176094617, 21.66468688315246326109397051216, 22.27958208435829124462158915198, 23.004969566368715001234684690594, 25.20176392969274830743388939316, 26.13946588245033991180024648025, 26.70020473789275943909385890089, 27.7486514402156506718157772591, 28.98312579605002562107272705976, 29.17638038944591895570499146658

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