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Properties

Label 1-113-113.15-r0-0-0
Degree $1$
Conductor $113$
Sign $0.910 - 0.413i$
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 − i·3-s + 4-s + i·5-s − i·6-s + 7-s + 8-s − 9-s + i·10-s − 11-s − i·12-s − 13-s + 14-s + 15-s + 16-s − i·17-s + ⋯
L(s)  = 1  + 2-s − i·3-s + 4-s + i·5-s − i·6-s + 7-s + 8-s − 9-s + i·10-s − 11-s − i·12-s − 13-s + 14-s + 15-s + 16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.910 - 0.413i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.910 - 0.413i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.812432691 - 0.3919783300i\)
\(L(\frac12)\) \(\approx\) \(1.812432691 - 0.3919783300i\)
\(L(1)\) \(\approx\) \(1.743837039 - 0.2774687268i\)
\(L(1)\) \(\approx\) \(1.743837039 - 0.2774687268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad113 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 + T \)
13 \( 1 + iT \)
17 \( 1 - iT \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + iT \)
37 \( 1 - T \)
41 \( 1 - iT \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 - T \)
71 \( 1 + iT \)
73 \( 1 + iT \)
79 \( 1 - iT \)
83 \( 1 - T \)
89 \( 1 - iT \)
97 \( 1 - iT \)
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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

−29.371102418008165381567415179000, −28.43151015388170996109885130035, −27.61043357271403889591865188045, −26.32372606537291572422945233511, −25.208400117174766220244498726397, −23.97996113772507800754673712734, −23.53690702954053310201083708539, −21.79488892298777915151583318896, −21.49366662602416080510358997001, −20.41701021023608832208473989833, −19.7484293154730246841112259310, −17.5300803440068968456224031010, −16.654598623730814304454914308466, −15.53919001031556160723352274737, −14.83396029247435310543809448319, −13.60116079547086733316360780601, −12.43108645248297583882457870230, −11.30986551627892369652006040510, −10.30156025152236745237469376722, −8.76838277735054512961471209077, −7.53953139054453592631025047593, −5.36414211781263789341618015016, −5.045309701738267421807077517259, −3.8079979231484882769417632805, −2.12062247882048841421218254360, 2.03889196536083433513607295452, 2.92987204007969302556329285475, 4.838833039584763371128501428603, 6.03375184924793451749622947459, 7.30540956784181647452220028041, 7.909804806288077771580958585013, 10.34008010666790557795649342984, 11.4214211029860129314110337234, 12.25144338607210933890002039915, 13.54261785677726636230142440110, 14.38154409416511413037056969022, 15.11002258011584394029707675492, 16.74360445610461320580156587971, 18.07109981419810098810366880289, 18.83770015507217525104787612391, 20.16542779851754686370115282645, 21.132437255531240317160474242513, 22.40933316314083919600209869080, 23.148472561290978687545767686605, 24.14480114751686101950822818301, 24.876745143433017989719980335526, 25.96042752291233516023765224805, 27.15021962984857992718130182879, 28.910117361643516509009843745285, 29.490381276110280999911442532672

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