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Properties

Label 1-1337-1337.13-r1-0-0
Degree $1$
Conductor $1337$
Sign $-0.481 - 0.876i$
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  + (0.922 − 0.386i)2-s + (0.724 − 0.689i)3-s + (0.701 − 0.712i)4-s + (0.986 − 0.164i)5-s + (0.401 − 0.915i)6-s + (0.371 − 0.928i)8-s + (0.0495 − 0.998i)9-s + (0.846 − 0.533i)10-s + (0.789 + 0.614i)11-s + (0.0165 − 0.999i)12-s + (−0.991 − 0.131i)13-s + (0.601 − 0.799i)15-s + (−0.0165 − 0.999i)16-s + (−0.115 + 0.993i)17-s + (−0.340 − 0.940i)18-s + (0.846 + 0.533i)19-s + ⋯
L(s)  = 1  + (0.922 − 0.386i)2-s + (0.724 − 0.689i)3-s + (0.701 − 0.712i)4-s + (0.986 − 0.164i)5-s + (0.401 − 0.915i)6-s + (0.371 − 0.928i)8-s + (0.0495 − 0.998i)9-s + (0.846 − 0.533i)10-s + (0.789 + 0.614i)11-s + (0.0165 − 0.999i)12-s + (−0.991 − 0.131i)13-s + (0.601 − 0.799i)15-s + (−0.0165 − 0.999i)16-s + (−0.115 + 0.993i)17-s + (−0.340 − 0.940i)18-s + (0.846 + 0.533i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1337\)    =    \(7 \cdot 191\)
Sign: $-0.481 - 0.876i$
Analytic conductor: \(143.680\)
Root analytic conductor: \(143.680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1337} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1337,\ (1:\ ),\ -0.481 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.718864541 - 6.289043921i\)
\(L(\frac12)\) \(\approx\) \(3.718864541 - 6.289043921i\)
\(L(1)\) \(\approx\) \(2.407692449 - 1.669628993i\)
\(L(1)\) \(\approx\) \(2.407692449 - 1.669628993i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.922 - 0.386i)T \)
3 \( 1 + (0.724 - 0.689i)T \)
5 \( 1 + (0.986 - 0.164i)T \)
11 \( 1 + (0.789 + 0.614i)T \)
13 \( 1 + (-0.991 - 0.131i)T \)
17 \( 1 + (-0.115 + 0.993i)T \)
19 \( 1 + (0.846 + 0.533i)T \)
23 \( 1 + (0.997 - 0.0660i)T \)
29 \( 1 + (-0.956 + 0.293i)T \)
31 \( 1 + (-0.546 - 0.837i)T \)
37 \( 1 + (0.546 - 0.837i)T \)
41 \( 1 + (0.0825 - 0.996i)T \)
43 \( 1 + (-0.213 - 0.976i)T \)
47 \( 1 + (0.999 + 0.0330i)T \)
53 \( 1 + (-0.340 + 0.940i)T \)
59 \( 1 + (0.627 - 0.778i)T \)
61 \( 1 + (0.909 - 0.416i)T \)
67 \( 1 + (0.980 + 0.197i)T \)
71 \( 1 + (-0.518 - 0.854i)T \)
73 \( 1 + (0.277 - 0.960i)T \)
79 \( 1 + (-0.956 - 0.293i)T \)
83 \( 1 + (-0.997 - 0.0660i)T \)
89 \( 1 + (-0.863 - 0.504i)T \)
97 \( 1 + (0.934 - 0.355i)T \)
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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

−21.18145409332179201071350177825, −20.37792894387583877847628080531, −19.81678164761073721533881752244, −18.815170724367435419009127628788, −17.696645401752900138214963267664, −16.83728375419231568168214686316, −16.39810701250069633864353252414, −15.452764565613136818375701786212, −14.54143003272513017139190296467, −14.31972520265084887119545773982, −13.45215179090997676826867626414, −12.93377286045365412249667482601, −11.59359657050695511726206205957, −11.10287459827341890969668772142, −9.86251135044079362916206352618, −9.32881605760517714937628361990, −8.47082213512228108241223516061, −7.29717052295501951069785025908, −6.76813157127853826112641807172, −5.50952644607773211973286570154, −5.05359088900644235876787373382, −4.10880910271272709151612370685, −2.97207115004160083800894074065, −2.65680752245137527941899981802, −1.39530482073045333933467901375, 0.83989454504083994455947344279, 1.83522931287327605797837668840, 2.2512887369437168649576610951, 3.364492480784282453306770982648, 4.1900014877972369279100386143, 5.34937165061909693509192660973, 6.013297579236891389093730788314, 6.99068458321311387898617430172, 7.50474809186081556480717996763, 8.96385320338198518937563578804, 9.53929880905079395115704160142, 10.301755234949438322309174396119, 11.396475110035728908714468947010, 12.4685657191009841798224571128, 12.67660896921638058468737477452, 13.52482402689300456370825744846, 14.39337146818110575833793346003, 14.65230413693648613908423404941, 15.49423556373952892325646201206, 16.87830436348319053049765229851, 17.347740464638776100043482492469, 18.462706937895906573167953459408, 19.095114707785312537582406285913, 19.989076502364621965198072864567, 20.42494869122568814530082003070

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