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Properties

Label 1-13-13.5-r1-0-0
Degree $1$
Conductor $13$
Sign $0.289 - 0.957i$
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  − i·2-s + 3-s − 4-s − i·5-s − i·6-s + i·7-s + i·8-s + 9-s − 10-s + i·11-s − 12-s + 14-s − i·15-s + 16-s − 17-s − i·18-s + ⋯
L(s)  = 1  − i·2-s + 3-s − 4-s − i·5-s − i·6-s + i·7-s + i·8-s + 9-s − 10-s + i·11-s − 12-s + 14-s − i·15-s + 16-s − 17-s − i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(13\)
Sign: $0.289 - 0.957i$
Analytic conductor: \(1.39704\)
Root analytic conductor: \(1.39704\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 13,\ (1:\ ),\ 0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080873679 - 0.8020687656i\)
\(L(\frac12)\) \(\approx\) \(1.080873679 - 0.8020687656i\)
\(L(1)\) \(\approx\) \(1.086429434 - 0.5814393878i\)
\(L(1)\) \(\approx\) \(1.086429434 - 0.5814393878i\)

Euler product

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

−43.66169372259601355647086681782, −42.33321828301002446414212889588, −42.01243734111834789179206571829, −39.99884476625807221373619354350, −37.94370316023537118571962014758, −36.74450646135960045266662861837, −35.42571280459602374366758944441, −33.80618812692548610046079809159, −32.61135344484786379077167851304, −31.2050934342588729157708559964, −29.87968724894081675819666253017, −26.91280133880953633992411090990, −26.45434783862766779809482397395, −24.95202266245560552819984002747, −23.4392965809666309731024672722, −21.7460382586681658260482353039, −19.55290227529063987885835251978, −18.10686380050982826089918877859, −16.0830899304950548429787234291, −14.468748219148105193603380433629, −13.587144687321544861182967302511, −10.21254084453347825369758588336, −8.206623201456027180127384654917, −6.7294196966943624248386542292, −3.7438215641461395319322782061, 2.19555319112541449367642087766, 4.56540124508568592878688473294, 8.48269853307877161078322980284, 9.59882957568404703438492303028, 12.14842691516461133225165992250, 13.39728080193893431404171908435, 15.34689368438853111429470458191, 17.91787476538898323746092421502, 19.59738667898703140158158422106, 20.59904703474526016023825974227, 21.93983499378017955326433862694, 24.21131436731631252498581961937, 25.8005879811408814363739761787, 27.61930936712106563830038196074, 28.713965931984609204490184782952, 30.62909489872992613680589947841, 31.5663026885561902954123742197, 32.6862225849004931411585046007, 35.50574902500719058128431674735, 36.53000016843034794854918904908, 37.76264121822764455968030883774, 38.88962429257123256447175596942, 40.54440810940620714296349107648, 41.709436176353847190567066381979, 43.76093287357656424939124511852

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