Properties

Label 1-113-113.13-r0-0-0
Degree $1$
Conductor $113$
Sign $-0.940 - 0.340i$
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  + (−0.623 + 0.781i)2-s + (0.330 + 0.943i)3-s + (−0.222 − 0.974i)4-s + (−0.330 + 0.943i)5-s + (−0.943 − 0.330i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (−0.781 + 0.623i)9-s + (−0.532 − 0.846i)10-s + (−0.974 − 0.222i)11-s + (0.846 − 0.532i)12-s + (−0.433 + 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (0.993 − 0.111i)17-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (0.330 + 0.943i)3-s + (−0.222 − 0.974i)4-s + (−0.330 + 0.943i)5-s + (−0.943 − 0.330i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (−0.781 + 0.623i)9-s + (−0.532 − 0.846i)10-s + (−0.974 − 0.222i)11-s + (0.846 − 0.532i)12-s + (−0.433 + 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (0.993 − 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $-0.940 - 0.340i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ -0.940 - 0.340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08028866876 + 0.4576198386i\)
\(L(\frac12)\) \(\approx\) \(-0.08028866876 + 0.4576198386i\)
\(L(1)\) \(\approx\) \(0.3804526868 + 0.4693711288i\)
\(L(1)\) \(\approx\) \(0.3804526868 + 0.4693711288i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad113 \( 1 \)
good2 \( 1 + (-0.623 + 0.781i)T \)
3 \( 1 + (0.330 + 0.943i)T \)
5 \( 1 + (-0.330 + 0.943i)T \)
7 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.974 - 0.222i)T \)
13 \( 1 + (-0.433 + 0.900i)T \)
17 \( 1 + (0.993 - 0.111i)T \)
19 \( 1 + (-0.943 + 0.330i)T \)
23 \( 1 + (0.943 + 0.330i)T \)
29 \( 1 + (-0.993 + 0.111i)T \)
31 \( 1 + (0.433 - 0.900i)T \)
37 \( 1 + (0.532 + 0.846i)T \)
41 \( 1 + (-0.974 + 0.222i)T \)
43 \( 1 + (0.111 + 0.993i)T \)
47 \( 1 + (0.846 + 0.532i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 + (0.943 - 0.330i)T \)
61 \( 1 + (0.974 + 0.222i)T \)
67 \( 1 + (-0.846 + 0.532i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (-0.532 + 0.846i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (-0.111 + 0.993i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−28.75633647808961094086169514042, −28.16164435110967719188007609581, −26.863601497565526143811820692530, −25.5633879449187219389654906868, −25.133011076431885759148147150323, −23.667435196907458976543935520552, −22.717260539352723101741141569594, −21.17843126947092920330584660713, −20.31110085910757421169473862956, −19.38404423531395210768406366218, −18.72784160439452943967414816437, −17.52701041464412620720479633183, −16.54789502340291474410593887625, −15.20743633897383715828806751208, −13.21070349518407330842711983993, −12.75985157816939352795590096415, −12.00242238418008882636925843596, −10.37982048970252845088690306189, −9.09702206469322149952504345931, −8.22534299923429012024920666837, −7.2048829228124353636924167368, −5.35652427657658312905815843115, −3.412016537131324187598971561150, −2.25238369765044463702786916785, −0.50641649341030994244022848258, 2.75499895429718495642106622042, 4.18853819713499077118240535878, 5.73662050611423354350398440177, 7.03608671510722730328175363088, 8.09580449431634128814451844978, 9.544010976341065159071731230343, 10.23998537260842198011506048657, 11.21901600524772936384654938648, 13.42142778175823003479785884498, 14.55279185354559743535036678262, 15.29617565963200803635910596156, 16.32930300611126660588859366235, 17.0443549285958486630861808916, 18.85458252228046909462839700597, 19.13862674826660623750125761982, 20.545115916781598738593912620959, 21.87239852115335617973548508256, 22.96763141470526956015855238411, 23.640186710662394640036758790053, 25.395748003959103934455993818806, 26.05246881228614713616017855433, 26.66714900667587212397385102546, 27.49743277459007830038260345072, 28.68242712011277866668025083862, 29.689699165890558454925392278219

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