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Properties

Label 1-113-113.69-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

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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

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

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: $\chi_{113} (69, \cdot )$
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 \( 1 + T \)
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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

−28.89142046932326005751772286159, −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