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Properties

Label 1-1343-1343.1300-r0-0-0
Degree $1$
Conductor $1343$
Sign $0.614 - 0.788i$
Analytic cond. $6.23686$
Root an. cond. $6.23686$
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  + (−0.0804 + 0.996i)2-s + (0.0201 − 0.999i)3-s + (−0.987 − 0.160i)4-s + (0.894 + 0.446i)5-s + (0.994 + 0.100i)6-s + (−0.482 − 0.875i)7-s + (0.239 − 0.970i)8-s + (−0.999 − 0.0402i)9-s + (−0.517 + 0.855i)10-s + (0.834 + 0.551i)11-s + (−0.180 + 0.983i)12-s + (0.632 − 0.774i)13-s + (0.911 − 0.410i)14-s + (0.464 − 0.885i)15-s + (0.948 + 0.316i)16-s + ⋯
L(s)  = 1  + (−0.0804 + 0.996i)2-s + (0.0201 − 0.999i)3-s + (−0.987 − 0.160i)4-s + (0.894 + 0.446i)5-s + (0.994 + 0.100i)6-s + (−0.482 − 0.875i)7-s + (0.239 − 0.970i)8-s + (−0.999 − 0.0402i)9-s + (−0.517 + 0.855i)10-s + (0.834 + 0.551i)11-s + (−0.180 + 0.983i)12-s + (0.632 − 0.774i)13-s + (0.911 − 0.410i)14-s + (0.464 − 0.885i)15-s + (0.948 + 0.316i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1343 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1343 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1343\)    =    \(17 \cdot 79\)
Sign: $0.614 - 0.788i$
Analytic conductor: \(6.23686\)
Root analytic conductor: \(6.23686\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1343} (1300, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1343,\ (0:\ ),\ 0.614 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.269381169 - 0.6200545243i\)
\(L(\frac12)\) \(\approx\) \(1.269381169 - 0.6200545243i\)
\(L(1)\) \(\approx\) \(1.058147997 + 0.02793952601i\)
\(L(1)\) \(\approx\) \(1.058147997 + 0.02793952601i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.0804 + 0.996i)T \)
3 \( 1 + (0.0201 - 0.999i)T \)
5 \( 1 + (0.894 + 0.446i)T \)
7 \( 1 + (-0.482 - 0.875i)T \)
11 \( 1 + (0.834 + 0.551i)T \)
13 \( 1 + (0.632 - 0.774i)T \)
19 \( 1 + (0.600 - 0.799i)T \)
23 \( 1 + (-0.258 - 0.965i)T \)
29 \( 1 + (-0.975 - 0.219i)T \)
31 \( 1 + (0.335 - 0.941i)T \)
37 \( 1 + (0.927 + 0.373i)T \)
41 \( 1 + (-0.998 + 0.0603i)T \)
43 \( 1 + (-0.979 - 0.200i)T \)
47 \( 1 + (0.919 + 0.391i)T \)
53 \( 1 + (-0.721 - 0.692i)T \)
59 \( 1 + (0.160 + 0.987i)T \)
61 \( 1 + (-0.616 - 0.787i)T \)
67 \( 1 + (0.568 + 0.822i)T \)
71 \( 1 + (0.517 + 0.855i)T \)
73 \( 1 + (0.100 - 0.994i)T \)
83 \( 1 + (0.160 - 0.987i)T \)
89 \( 1 + (0.970 - 0.239i)T \)
97 \( 1 + (0.787 - 0.616i)T \)
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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

−21.19075572320515573819718809817, −20.32846899671929177245027059093, −19.73421520174907109164460261732, −18.75570848307848374550745637407, −18.17096365564467300259153105594, −17.10757084438488118720963241028, −16.614263914573881099058328414186, −15.83189904431630276696344541092, −14.69286301753416085788975233778, −13.9692824944356182842575925391, −13.42112707440614295281970610714, −12.28932502964170253094850170360, −11.707063111333144720430028751494, −10.93118173264114209083746177620, −9.95392265960587724443713291459, −9.35343849053286014811444794591, −8.98977192278176396267595055996, −8.19109279635962539176372056458, −6.343560256404151403403603786407, −5.64166483898434172236343701978, −4.96140857968420254999574539319, −3.800781295239813962566214806650, −3.29785587976284233875470032163, −2.152136141359554517952029609769, −1.28180844519444987944117682232, 0.617672605128984482142067919604, 1.571273543206932902868506274952, 2.87715796398394620319865966801, 3.873488294351115168501375534367, 5.0724996231585038957775010348, 6.104411026644709350067533312155, 6.50453801686649246485473428484, 7.21785472104137226239798109933, 7.93181969445759978909980002525, 8.97515594132823243658477484163, 9.70405027326239812561005267578, 10.49975926152424157178517392026, 11.56303196818474940810698432548, 12.83230578837147643627728484302, 13.26538228719374452069613466687, 13.855765575957161824308459082870, 14.57966945330339717628094913525, 15.30382975845978627612522310081, 16.52381592509188067417112818968, 17.134530662708376686415789425709, 17.61590192689429239046326921735, 18.40178318974175389036052227069, 18.906852728056256252224268479438, 20.03451406089380466954072271778, 20.47809067884043633380677663019

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