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Properties

Label 1-103-103.56-r0-0-0
Degree $1$
Conductor $103$
Sign $0.957 + 0.287i$
Analytic cond. $0.478329$
Root an. cond. $0.478329$
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  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + 9-s + 10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + 9-s + 10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.957 + 0.287i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.957 + 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9121643839 + 0.1340828719i\)
\(L(\frac12)\) \(\approx\) \(0.9121643839 + 0.1340828719i\)
\(L(1)\) \(\approx\) \(0.9566360406 + 0.03046771821i\)
\(L(1)\) \(\approx\) \(0.9566360406 + 0.03046771821i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−29.66035142289358948855023964397, −28.5416129969883256039776043008, −27.19230953188873784087426181000, −26.73251534045560103088120254004, −25.60752632295455321397990833319, −24.76781075360688001513169231164, −23.78313058537364727731714576288, −22.99387936567413185219712776384, −21.05807161239827994223901518302, −20.12554285532360323058299827557, −19.238604723282277606906622553927, −18.29342963528112424409906869345, −16.56604467814664094914424154756, −16.14636965404429979568997462640, −14.99373896121737460736067690813, −13.60320954314349915615913235177, −13.128645427678010508649368773035, −10.91888989806939004017245212144, −9.59757597894682078171200461030, −8.57002702936793181211017571735, −7.82992277824062869415002159765, −6.540900494854239054166616051999, −4.79256517985561482222065871792, −3.52119987678888935467758711776, −1.097632909709277907250057242554, 2.15251402654076763500940749306, 3.06020855046176389211898025637, 4.265034285751608326704199506548, 6.74228422596651221852016677749, 8.03585868203078107504622468789, 8.98230268881127331632896614537, 10.12531422877474170851070460948, 11.214041414333496444819211997477, 12.62050672451647759488186742050, 13.44934015929058263774534446775, 15.06614119777492985514753534685, 15.6982089010745527615348235271, 17.63512067075010348360360964827, 18.76126034847796594896808540023, 19.17569008380694330438324911222, 20.30147527620929564693024826975, 21.32636621073206264682403310541, 22.272034243727688115458281939807, 23.43660854028889132612019257846, 25.261209178059969427293051322500, 25.95487478660274483370732825468, 26.63249417467365840597682854987, 27.88355605709438817875850141673, 28.68059952455716184548441629493, 30.190841069748228948122552125595

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