# Properties

 Label 1-13-13.8-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 $0$

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## 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: $\chi_{13} (8, \cdot )$ 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

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