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Properties

Label 1-2443-2443.1395-r0-0-0
Degree $1$
Conductor $2443$
Sign $-0.895 + 0.444i$
Analytic cond. $11.3452$
Root an. cond. $11.3452$
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 + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 6-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s − 13-s + 15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 6-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s − 13-s + 15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2443 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2443 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2443\)    =    \(7 \cdot 349\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(11.3452\)
Root analytic conductor: \(11.3452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2443} (1395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2443,\ (0:\ ),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1709164503 + 0.7295975221i\)
\(L(\frac12)\) \(\approx\) \(0.1709164503 + 0.7295975221i\)
\(L(1)\) \(\approx\) \(0.6702791467 + 0.4563194791i\)
\(L(1)\) \(\approx\) \(0.6702791467 + 0.4563194791i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
349 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 - T \)
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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

−19.25255730823352588128619134486, −18.63382465407055806665540940288, −17.929719088585779669873572199867, −17.44876250551253987444107746699, −16.40878255258302202968888357869, −15.29436401702862524605179556865, −14.76768272532210800347310210353, −14.01489663127915629347968245678, −13.417843670661704655917590302959, −12.446517053586058288804998028938, −11.862924980684081988310648250256, −11.636301343563165183070014717844, −10.56391921680206559789799095949, −10.075638618946530468176846966566, −9.148535260300239981844687026710, −7.972192444331654539004883856649, −7.228734490687147635143295580547, −6.58207099121730247233709670617, −5.79273550664406632001284594351, −4.82325795339638994321685856706, −4.14363017188047043616986682243, −3.08168759920691854544974645358, −2.2801896635858352867164307782, −1.67610737847927677233579996266, −0.30916401046834718579839454222, 0.7411862995765673488894725129, 2.53863270138199406691268412240, 3.63907897587741365256694161653, 4.228439477292747609879855553236, 4.840739287041610886133157178327, 5.53230871644494264355298978577, 6.35476836853312467715131849523, 7.057603142620418310992976737804, 8.257226496194802969978815813577, 8.71665838350176916480897370422, 9.31047448042141328747710548095, 10.383903244695954767639490660819, 11.26271761203797174130199133338, 12.04215028166487391946939687446, 12.551003156311825643738246170058, 13.34270526078438165815797748572, 14.46373245112901452748047741138, 14.76224613087927179428561130817, 15.83079144880259909710655214240, 16.06206084911265219803344298191, 16.81812017612337282935123421430, 17.37139482180657129149316860331, 17.9002607306703865398625489884, 19.32925921895152359493245367512, 19.7754505185283599406372238300

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