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Properties

Label 1-2681-2681.103-r1-0-0
Degree $1$
Conductor $2681$
Sign $0.00926 - 0.999i$
Analytic cond. $288.113$
Root an. cond. $288.113$
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  + (−0.997 + 0.0712i)2-s + (0.878 + 0.478i)3-s + (0.989 − 0.142i)4-s + (0.0630 + 0.998i)5-s + (−0.909 − 0.414i)6-s + (−0.977 + 0.212i)8-s + (0.542 + 0.840i)9-s + (−0.133 − 0.990i)10-s + (−0.907 − 0.419i)11-s + (0.937 + 0.348i)12-s + (0.692 − 0.721i)13-s + (−0.422 + 0.906i)15-s + (0.959 − 0.281i)16-s + (−0.112 + 0.993i)17-s + (−0.600 − 0.799i)18-s + (−0.336 − 0.941i)19-s + ⋯
L(s)  = 1  + (−0.997 + 0.0712i)2-s + (0.878 + 0.478i)3-s + (0.989 − 0.142i)4-s + (0.0630 + 0.998i)5-s + (−0.909 − 0.414i)6-s + (−0.977 + 0.212i)8-s + (0.542 + 0.840i)9-s + (−0.133 − 0.990i)10-s + (−0.907 − 0.419i)11-s + (0.937 + 0.348i)12-s + (0.692 − 0.721i)13-s + (−0.422 + 0.906i)15-s + (0.959 − 0.281i)16-s + (−0.112 + 0.993i)17-s + (−0.600 − 0.799i)18-s + (−0.336 − 0.941i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2681 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00926 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2681 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00926 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2681\)    =    \(7 \cdot 383\)
Sign: $0.00926 - 0.999i$
Analytic conductor: \(288.113\)
Root analytic conductor: \(288.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2681} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2681,\ (1:\ ),\ 0.00926 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2098831068 - 0.2079469585i\)
\(L(\frac12)\) \(\approx\) \(0.2098831068 - 0.2079469585i\)
\(L(1)\) \(\approx\) \(0.7713022288 + 0.2696580901i\)
\(L(1)\) \(\approx\) \(0.7713022288 + 0.2696580901i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
383 \( 1 \)
good2 \( 1 + (-0.997 + 0.0712i)T \)
3 \( 1 + (0.878 + 0.478i)T \)
5 \( 1 + (0.0630 + 0.998i)T \)
11 \( 1 + (-0.907 - 0.419i)T \)
13 \( 1 + (0.692 - 0.721i)T \)
17 \( 1 + (-0.112 + 0.993i)T \)
19 \( 1 + (-0.336 - 0.941i)T \)
23 \( 1 + (-0.320 - 0.947i)T \)
29 \( 1 + (-0.330 - 0.943i)T \)
31 \( 1 + (0.432 + 0.901i)T \)
37 \( 1 + (0.622 + 0.782i)T \)
41 \( 1 + (-0.551 + 0.834i)T \)
43 \( 1 + (0.578 + 0.815i)T \)
47 \( 1 + (0.225 + 0.974i)T \)
53 \( 1 + (-0.246 - 0.969i)T \)
59 \( 1 + (-0.971 + 0.238i)T \)
61 \( 1 + (-0.978 - 0.206i)T \)
67 \( 1 + (0.273 - 0.961i)T \)
71 \( 1 + (0.864 + 0.502i)T \)
73 \( 1 + (0.893 + 0.449i)T \)
79 \( 1 + (-0.994 + 0.103i)T \)
83 \( 1 + (-0.973 - 0.228i)T \)
89 \( 1 + (-0.0684 + 0.997i)T \)
97 \( 1 + (0.902 - 0.429i)T \)
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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

−19.226219896505383760469419659912, −18.48133073346174681238399015627, −18.2246448790061337939199926171, −17.22059433478942262872733485631, −16.57446438091030984861181161649, −15.692972096680569921012942664485, −15.46379095456208218656296903306, −14.248787240940052646236648037975, −13.58255918124883291195158938926, −12.78946568838923124973114041973, −12.18804651193500876544150243607, −11.48106927276074158819441461717, −10.42725122187308689666349458559, −9.643429741304241798373537903443, −9.07027531956292462828518117397, −8.54280304360142856903934090832, −7.66288211437931823729397446690, −7.34927436233576112377981262259, −6.23266772512197685662447781923, −5.44374737608665035418127355739, −4.21899665305589687732219860030, −3.41650508883393631973879815893, −2.29468183372224260487173578425, −1.783593250581947111305642552273, −0.92898443577758410428396553195, 0.06063680028097741758625882016, 1.375793598362566159085237253462, 2.48518343781343143839870115706, 2.84578553984846670204198212069, 3.66038705323702303824124990993, 4.80853989661930143225413155681, 6.03752854632176316799271099410, 6.51257265083039196070225123846, 7.62601145153112486394419084535, 8.12067721189041442361508167967, 8.632427684255914561335053844018, 9.66811214605900118664306171242, 10.23279524998153048142949231482, 10.875335653480797102675695737, 11.20683194606278665737194961288, 12.62037381668679624112908885897, 13.34008350439679917368875931887, 14.186361007993545288463713725876, 14.97479721832676894165307253026, 15.48613079885268130568827805609, 15.901868874120861033498517978258, 16.891857393828174429426892652221, 17.68269380101156738227094324810, 18.48731606955741393820516600945, 18.77618800777870709707155653977

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