Properties

Degree 1
Conductor 113
Sign $0.473 + 0.880i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.473 + 0.880i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.473 + 0.880i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(113\)
\( \varepsilon \)  =  $0.473 + 0.880i$
motivic weight  =  \(0\)
character  :  $\chi_{113} (69, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 113,\ (0:\ ),\ 0.473 + 0.880i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5798294569 + 0.3464780233i$
$L(\frac12,\chi)$  $\approx$  $0.5798294569 + 0.3464780233i$
$L(\chi,1)$  $\approx$  0.6659895756 + 0.2248482376i
$L(1,\chi)$  $\approx$  0.6659895756 + 0.2248482376i

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

−28.89142046932326005751772286159, −28.13818496912123785944338723540, −27.63262183687308332022579930613, −26.046372901862647057115320381439, −24.94072172331858797517885133282, −24.47087472681209800630476359664, −23.374914033210701487395784079072, −21.8475089236940585670105920942, −20.6707382172637404276984766080, −19.87569759103652713497599644130, −18.37719337985653711629243601120, −17.64139862990759334158910401141, −17.19065892485599590367669999899, −15.931002958066589132377037800362, −14.48241558554541501684284012477, −12.86553997761745496293223320156, −12.05415359679773647541319465735, −10.81215016092204567652695267959, −9.80208324027894358069470164524, −8.30606029458129421396275166903, −7.487947836731803753844987903406, −6.03391907061096542779353625257, −5.00363366118806060912040175757, −2.18588129344979785388093366864, −1.15732128404212423630566029663, 1.56980204978341814053179473901, 3.36495498894692313607572228912, 5.37070531239024846018585599277, 6.35005572731971566649562236760, 7.74372555848332031129854568730, 9.25267740512702377881272297780, 10.081975314330350194225199557444, 11.267338862115988396725861649477, 11.69087563769346470393268111407, 14.01174211816705642754092673661, 14.96427863660577899707882582622, 16.34411088546891974825108820251, 16.98028567058654205998537302101, 18.19615276625620095884168226428, 18.63789881583161816182003186268, 20.443977141765264311781049123706, 21.32139568636241479474989083133, 21.978779132562983054671684514549, 23.57108691550580296663348538842, 24.54317419820919263041451639173, 25.82701069616020835537329675756, 26.77019529028705527028789271201, 27.32795395759900667418754666555, 28.42638528636722324552196321131, 29.40269175370707057111937169314

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