# Properties

 Label 1-967-967.646-r1-0-0 Degree $1$ Conductor $967$ Sign $-0.946 + 0.323i$ Analytic cond. $103.918$ Root an. cond. $103.918$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.538 − 0.842i)2-s + (−0.682 + 0.730i)3-s + (−0.419 − 0.907i)4-s + (−0.983 − 0.181i)5-s + (0.247 + 0.968i)6-s + (−0.934 + 0.356i)7-s + (−0.990 − 0.136i)8-s + (−0.0682 − 0.997i)9-s + (−0.682 + 0.730i)10-s + (−0.0682 − 0.997i)11-s + (0.949 + 0.313i)12-s + (0.829 + 0.557i)13-s + (−0.203 + 0.979i)14-s + (0.803 − 0.595i)15-s + (−0.648 + 0.761i)16-s + (−0.917 + 0.398i)17-s + ⋯
 L(s)  = 1 + (0.538 − 0.842i)2-s + (−0.682 + 0.730i)3-s + (−0.419 − 0.907i)4-s + (−0.983 − 0.181i)5-s + (0.247 + 0.968i)6-s + (−0.934 + 0.356i)7-s + (−0.990 − 0.136i)8-s + (−0.0682 − 0.997i)9-s + (−0.682 + 0.730i)10-s + (−0.0682 − 0.997i)11-s + (0.949 + 0.313i)12-s + (0.829 + 0.557i)13-s + (−0.203 + 0.979i)14-s + (0.803 − 0.595i)15-s + (−0.648 + 0.761i)16-s + (−0.917 + 0.398i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$967$$ Sign: $-0.946 + 0.323i$ Analytic conductor: $$103.918$$ Root analytic conductor: $$103.918$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{967} (646, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 967,\ (1:\ ),\ -0.946 + 0.323i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$-0.09478708591 - 0.5693280851i$$ $$L(\frac12)$$ $$\approx$$ $$-0.09478708591 - 0.5693280851i$$ $$L(1)$$ $$\approx$$ $$0.6778871032 - 0.3262500010i$$ $$L(1)$$ $$\approx$$ $$0.6778871032 - 0.3262500010i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad967 $$1$$
good2 $$1 + (-0.538 + 0.842i)T$$
3 $$1 + (0.682 - 0.730i)T$$
5 $$1 + (0.983 + 0.181i)T$$
7 $$1 + (0.934 - 0.356i)T$$
11 $$1 + (0.0682 + 0.997i)T$$
13 $$1 + (-0.829 - 0.557i)T$$
17 $$1 + (0.917 - 0.398i)T$$
19 $$1 + (-0.974 - 0.225i)T$$
23 $$1 + (-0.576 + 0.816i)T$$
29 $$1 + (-0.0682 + 0.997i)T$$
31 $$1 + (-0.613 + 0.789i)T$$
37 $$1 + (-0.829 - 0.557i)T$$
41 $$1 + (0.962 + 0.269i)T$$
43 $$1 + (0.113 + 0.993i)T$$
47 $$1 + (0.803 - 0.595i)T$$
53 $$1 + (-0.995 - 0.0909i)T$$
59 $$1 + (-0.995 + 0.0909i)T$$
61 $$1 + (0.715 - 0.699i)T$$
67 $$1 + (-0.990 + 0.136i)T$$
71 $$1 + (-0.460 + 0.887i)T$$
73 $$1 + (-0.995 + 0.0909i)T$$
79 $$1 + (0.995 - 0.0909i)T$$
83 $$1 + (0.247 + 0.968i)T$$
89 $$1 + (0.377 - 0.926i)T$$
97 $$1 - T$$
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−22.50965571488449066667794017819, −21.57145907038686031891267143306, −20.10365472053676130216009133769, −19.8136931299753791631767250668, −18.48554686857001957217946231153, −18.04395707023771885388042601912, −17.2094333002799268225146030210, −16.1755563741352738650551301914, −15.864343884133721029541140317557, −15.05051689655283291682765838898, −13.88456832609202694090674417862, −13.10570194757776559108649721023, −12.651333243390328223429395170873, −11.73063509689119947621409318146, −11.05018904604270615434821441437, −9.82075831643577441941928635493, −8.6304629182123791695609602472, −7.729055486648418820977465975988, −6.924683003218269679112666729792, −6.70358061742334263442155370618, −5.43534148588244623027506908954, −4.67925705675510144831661602567, −3.60900464863575977753061949106, −2.77802211760032205128639197888, −0.96530514868887263549477603677, 0.17219171666253684505729206236, 0.93420673203333125895047616374, 2.70574993698272016760823596904, 3.604388073920002810430630081253, 4.11004569836189010367760577977, 5.10312763549049830929171191417, 6.081923497383456398489158033478, 6.64091273055065684254724717732, 8.43585992948905786112872844296, 9.072977616209296919838854763253, 9.98292668735444200185928248300, 10.880667204976812196177981295768, 11.52497149725382462417414183591, 12.00378358440376185855912344341, 13.02198074101404283229591107162, 13.70030283907508794687678139973, 14.978526521891189156438297127, 15.575898250158014303469881465088, 16.181580147357180590344629382386, 16.96206616718751675724849872129, 18.40172136853703568283710680443, 18.80635885900722501420959152982, 19.669504032008462847804108166068, 20.4750553301777102324566071923, 21.1604055820334594818491330697