↑

Properties

Label 1-113-113.95-r0-0-0
Degree $1$
Conductor $113$
Sign $0.473 - 0.880i$
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

Related objects

Downloads

Learn more

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

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

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.473 - 0.880i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.473 - 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5798294569 - 0.3464780233i\)
\(L(\frac12)\) \(\approx\) \(0.5798294569 - 0.3464780233i\)
\(L(1)\) \(\approx\) \(0.6659895756 - 0.2248482376i\)
\(L(1)\) \(\approx\) \(0.6659895756 - 0.2248482376i\)

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 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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