Properties

Label 1-13-13.11-r1-0-0
Degree $1$
Conductor $13$
Sign $-0.0386 + 0.999i$
Analytic cond. $1.39704$
Root an. cond. $1.39704$
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  + (−0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + i·5-s + (0.866 − 0.5i)6-s + (−0.866 + 0.5i)7-s i·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.866 + 0.5i)11-s − 12-s + 14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + i·18-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + i·5-s + (0.866 − 0.5i)6-s + (−0.866 + 0.5i)7-s i·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.866 + 0.5i)11-s − 12-s + 14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0386 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0386 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(13\)
Sign: $-0.0386 + 0.999i$
Analytic conductor: \(1.39704\)
Root analytic conductor: \(1.39704\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 13,\ (1:\ ),\ -0.0386 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4048426516 + 0.4207849217i\)
\(L(\frac12)\) \(\approx\) \(0.4048426516 + 0.4207849217i\)
\(L(1)\) \(\approx\) \(0.5541243135 + 0.2274472859i\)
\(L(1)\) \(\approx\) \(0.5541243135 + 0.2274472859i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + iT \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + iT \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 + T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + iT \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−43.31869206681314121024218697364, −42.16399981927434131293910640324, −40.68125181149182130272470469685, −39.153411392899183147832475599836, −37.135826824282527664754407469293, −35.72998640308633489947819447472, −35.35326744678245210773092578721, −33.53515048617097616610864407062, −32.10639849816909610878347145286, −29.65859842980138898695042621752, −28.736645568233990207234910851386, −27.32094051196351555897323075834, −25.344031289518070239768624122379, −24.345665661610536048296495517831, −22.960773794814462804329085667052, −20.11901294655637634170559502532, −18.9095534244657775320001215005, −17.143808026292951051401463673262, −16.288710789877748904128325131, −13.62042425383954721789050127619, −11.706748031945682213428198890216, −9.4371452978705126240794561274, −7.54256479121809474504333672637, −5.83466878916868584082119055999, −0.88396030950993018705471470283, 3.32983235922570662743605525822, 6.61197886017718867260327598631, 9.304704200469930870841748121573, 10.561822846541455454688023478973, 12.08823121485062339199926124567, 15.125320499926784417604909240178, 16.675390801915910727107958823348, 18.22982679115041646341468567056, 19.79230458416173309976920824676, 21.68345121612472102029273094980, 22.606637778127120883138212077420, 25.54504856483978748166742087747, 26.62934932740005227578451524858, 27.96493575941531813729777014432, 29.16624987037204621005798590892, 30.7070180131503167341953669036, 32.8025496744660509150217686967, 34.38397138773570447099563880522, 35.26735320232653865270404415406, 37.38019130994940027036686271568, 38.405559501902477690422863287, 39.16084978067687771532204218338, 41.08433833151681631208230972752, 43.08803950439277744537810021651, 44.47443103977634728545471222112

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