Properties

Label 1-13-13.2-r1-0-0
Degree $1$
Conductor $13$
Sign $0.522 + 0.852i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(13\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(1.39704\)
Root analytic conductor: \(1.39704\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 13,\ (1:\ ),\ 0.522 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346813704 + 0.7545968286i\)
\(L(\frac12)\) \(\approx\) \(1.346813704 + 0.7545968286i\)
\(L(1)\) \(\approx\) \(1.300817583 + 0.5339369191i\)
\(L(1)\) \(\approx\) \(1.300817583 + 0.5339369191i\)

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

−42.916493459784617707701619768426, −41.575207017812764885467506475328, −40.884849005313018622653349190751, −39.446147257535379165436770301338, −37.98662174149648259326393740307, −36.59736365804954566584311360769, −34.36362583119418884864240426704, −33.73503213528638777331012671536, −31.3714238119598700960972215137, −30.44161532015156837717137813753, −29.31249399810574252421122238580, −27.82848702324585128669644429138, −25.307477164656216181300697786524, −23.7723130738260825346680194365, −22.707233933565961819535493352100, −21.193797014928129437987555228002, −19.12658986031807858311542402985, −17.95727240089126309718459523372, −15.19982307397033042107583736742, −13.68260616414035954960946604229, −11.976074282559220921024912566493, −10.74269657978003853822168308889, −7.21770260402982876163191488360, −5.37688302580705465622725578544, −2.34546853359506404994450241032, 4.24460934264584967159571245224, 5.57713196951261458498668689850, 8.28909241938397139268805863170, 10.93323919116728175068762537924, 12.71207036592429069533654996003, 14.662973932502194903070243897304, 16.25589362511344762130165698068, 17.25138541491035773839727275862, 20.65426041658851831825796728705, 21.36664226902833192750943599219, 23.25372235027411254080364224307, 24.26191061336298593641419872381, 26.22481109400446508001175625025, 27.78380944627266683281517490520, 29.444571537950115649748331238947, 31.38841268898555875278187245581, 32.53738083646613564275504133147, 33.61489336611419017475205563412, 34.84394463394565934266337377085, 36.81290632835271241146079108369, 38.90716595030295196315185881850, 39.85967535328406576141814214294, 40.70096083791469259116137271408, 42.76588365433364961049527879308, 43.838348329304751474850602875092

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