Properties

Degree 1
Conductor 113
Sign $0.294 - 0.955i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.294 - 0.955i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.294 - 0.955i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(113\)
\( \varepsilon \)  =  $0.294 - 0.955i$
motivic weight  =  \(0\)
character  :  $\chi_{113} (61, \cdot )$
Sato-Tate  :  $\mu(56)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 113,\ (0:\ ),\ 0.294 - 0.955i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.511707683 - 1.116050036i$
$L(\frac12,\chi)$  $\approx$  $1.511707683 - 1.116050036i$
$L(\chi,1)$  $\approx$  1.611864444 - 0.7827682471i
$L(1,\chi)$  $\approx$  1.611864444 - 0.7827682471i

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

−30.20771338444463441681271502294, −28.66539171965070474995619561931, −26.952306922085208147277001005953, −26.442775214161552681679843958078, −25.69446289116629189511953116707, −24.261452750426964860939169334262, −23.473019675493777012645318383486, −22.44931779388682893524395999889, −21.46064229176478990224886749924, −20.28351860565863497112024983144, −19.724453805588692812056257376257, −18.16337887653743201912383963417, −16.3393495999812259915731895322, −15.837869060632016041002661094323, −14.8410640211883314491028383255, −13.74660136767057797135523701576, −13.076653547062759025953909329357, −11.23109399872474560564184874670, −10.52401204827705389088672880321, −8.50838604532414536912766307026, −7.677999067888902323314675119827, −6.49868828420890300825186606414, −4.64664011951541139013715954267, −3.7046608917975419179171485411, −2.77922858734708810303487635318, 1.71782006758190474231162670815, 3.03641759551238842171180071558, 4.212823249086902425937344256259, 5.69183621734883906475560873828, 7.15776563750538245910779705395, 8.4186856707592477962379246908, 9.61489425451469073046936044099, 11.360239869545112086231668567261, 12.36961585847596123191794304885, 13.046214303522159540373061380084, 14.20654391047094319533166560659, 15.59187822009674776148540817807, 15.76358758894604164210081454670, 18.24850077399136026531906415585, 18.99130238787649231980557603399, 20.15781428364994250282948803177, 20.64881851727550538691496198739, 21.89709439492458193024257761759, 23.25687018138749625569071594241, 23.89633307337782950675892861635, 24.95522285691956996102132323046, 25.716725727050282989340010034754, 27.30058282858857388055940182992, 28.49580333160493804633301629755, 29.15509660978582522549397314478

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