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Properties

Label 1-1337-1337.789-r1-0-0
Degree $1$
Conductor $1337$
Sign $0.314 - 0.949i$
Analytic cond. $143.680$
Root an. cond. $143.680$
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  + (0.451 − 0.892i)2-s + (0.754 − 0.656i)3-s + (−0.592 − 0.805i)4-s + (−0.993 + 0.110i)5-s + (−0.245 − 0.969i)6-s + (−0.986 + 0.164i)8-s + (0.137 − 0.990i)9-s + (−0.350 + 0.936i)10-s + (0.904 + 0.426i)11-s + (−0.975 − 0.218i)12-s + (−0.945 + 0.324i)13-s + (−0.677 + 0.735i)15-s + (−0.298 + 0.954i)16-s + (−0.851 + 0.523i)17-s + (−0.821 − 0.569i)18-s + (−0.350 − 0.936i)19-s + ⋯
L(s)  = 1  + (0.451 − 0.892i)2-s + (0.754 − 0.656i)3-s + (−0.592 − 0.805i)4-s + (−0.993 + 0.110i)5-s + (−0.245 − 0.969i)6-s + (−0.986 + 0.164i)8-s + (0.137 − 0.990i)9-s + (−0.350 + 0.936i)10-s + (0.904 + 0.426i)11-s + (−0.975 − 0.218i)12-s + (−0.945 + 0.324i)13-s + (−0.677 + 0.735i)15-s + (−0.298 + 0.954i)16-s + (−0.851 + 0.523i)17-s + (−0.821 − 0.569i)18-s + (−0.350 − 0.936i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1337\)    =    \(7 \cdot 191\)
Sign: $0.314 - 0.949i$
Analytic conductor: \(143.680\)
Root analytic conductor: \(143.680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1337} (789, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1337,\ (1:\ ),\ 0.314 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.665343160 - 1.202029600i\)
\(L(\frac12)\) \(\approx\) \(1.665343160 - 1.202029600i\)
\(L(1)\) \(\approx\) \(0.9835483603 - 0.7502339130i\)
\(L(1)\) \(\approx\) \(0.9835483603 - 0.7502339130i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.451 - 0.892i)T \)
3 \( 1 + (0.754 - 0.656i)T \)
5 \( 1 + (-0.993 + 0.110i)T \)
11 \( 1 + (0.904 + 0.426i)T \)
13 \( 1 + (-0.945 + 0.324i)T \)
17 \( 1 + (-0.851 + 0.523i)T \)
19 \( 1 + (-0.350 - 0.936i)T \)
23 \( 1 + (0.350 + 0.936i)T \)
29 \( 1 + (-0.677 + 0.735i)T \)
31 \( 1 + (0.926 - 0.376i)T \)
37 \( 1 + (-0.926 - 0.376i)T \)
41 \( 1 + (-0.546 + 0.837i)T \)
43 \( 1 + (0.245 + 0.969i)T \)
47 \( 1 + (0.821 - 0.569i)T \)
53 \( 1 + (0.904 + 0.426i)T \)
59 \( 1 + (0.926 - 0.376i)T \)
61 \( 1 + (-0.851 - 0.523i)T \)
67 \( 1 + (0.0275 - 0.999i)T \)
71 \( 1 + (0.546 - 0.837i)T \)
73 \( 1 + (0.821 + 0.569i)T \)
79 \( 1 + (-0.298 + 0.954i)T \)
83 \( 1 + (0.986 - 0.164i)T \)
89 \( 1 + (0.962 - 0.272i)T \)
97 \( 1 + (-0.789 + 0.614i)T \)
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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

−20.76845064190637746331457175519, −20.34608839284462068858364025229, −19.210689523135848176642547656021, −18.91646067392462418745366643976, −17.50790882388178621716288530519, −16.751210574868479207368032710115, −16.21926949234427228977073161471, −15.30025336396044828552020913572, −14.993305091038963401688637243248, −14.16111965264247358856970327119, −13.51872838349395291477248028121, −12.412982202861979606455133319849, −11.859466104833336395060959582427, −10.74539628329839897118587215841, −9.73149738761843975161991005401, −8.69792744072570364889121862703, −8.46754646964386316857468702179, −7.41507549086079412668698672187, −6.83221057001726718620489564610, −5.57610833221005597168601811478, −4.6241220942084617567133200797, −4.0833740059288520635488201627, −3.33214233690579348874577552784, −2.366848829947210499928020377780, −0.446784268473785463478152672569, 0.685776764381994124470406418333, 1.75157476119753000462956163596, 2.55668235078142384751460879078, 3.513209896574154540290625971103, 4.17464368436753267343753401587, 4.99788847800607390829777574030, 6.50783081684340751513574429366, 7.02307879812665634826663949161, 8.05904879861635909575346473076, 9.03386077155151987719754459505, 9.44184139117331578564237054995, 10.67954748697223082368158356456, 11.55264387008919305899764778229, 12.08161110774009248466512092641, 12.79390339958446830177412480471, 13.53646583139837899289883922927, 14.395466528140931526051818524683, 15.09413637672814031427396995402, 15.43441188788167170045513640075, 17.028123269436645069326811952893, 17.75210617363174552253744356141, 18.67511728661276628643337610504, 19.448903104139461536518155650241, 19.725649297254813751636883673324, 20.21334730592783827652504301729

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