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Properties

Label 1-1580-1580.239-r1-0-0
Degree $1$
Conductor $1580$
Sign $-0.662 + 0.749i$
Analytic cond. $169.794$
Root an. cond. $169.794$
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  + (0.948 + 0.316i)3-s + (−0.200 − 0.979i)7-s + (0.799 + 0.600i)9-s + (0.996 − 0.0804i)11-s + (0.0402 + 0.999i)13-s + (−0.885 − 0.464i)17-s + (0.632 + 0.774i)19-s + (0.120 − 0.992i)21-s + (−0.5 + 0.866i)23-s + (0.568 + 0.822i)27-s + (−0.919 − 0.391i)29-s + (−0.692 + 0.721i)31-s + (0.970 + 0.239i)33-s + (−0.987 + 0.160i)37-s + (−0.278 + 0.960i)39-s + ⋯
L(s)  = 1  + (0.948 + 0.316i)3-s + (−0.200 − 0.979i)7-s + (0.799 + 0.600i)9-s + (0.996 − 0.0804i)11-s + (0.0402 + 0.999i)13-s + (−0.885 − 0.464i)17-s + (0.632 + 0.774i)19-s + (0.120 − 0.992i)21-s + (−0.5 + 0.866i)23-s + (0.568 + 0.822i)27-s + (−0.919 − 0.391i)29-s + (−0.692 + 0.721i)31-s + (0.970 + 0.239i)33-s + (−0.987 + 0.160i)37-s + (−0.278 + 0.960i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1580\)    =    \(2^{2} \cdot 5 \cdot 79\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(169.794\)
Root analytic conductor: \(169.794\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1580} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1580,\ (1:\ ),\ -0.662 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7458528618 + 1.654898532i\)
\(L(\frac12)\) \(\approx\) \(0.7458528618 + 1.654898532i\)
\(L(1)\) \(\approx\) \(1.310542460 + 0.2545199957i\)
\(L(1)\) \(\approx\) \(1.310542460 + 0.2545199957i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
79 \( 1 \)
good3 \( 1 + (0.948 + 0.316i)T \)
7 \( 1 + (-0.200 - 0.979i)T \)
11 \( 1 + (0.996 - 0.0804i)T \)
13 \( 1 + (0.0402 + 0.999i)T \)
17 \( 1 + (-0.885 - 0.464i)T \)
19 \( 1 + (0.632 + 0.774i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.919 - 0.391i)T \)
31 \( 1 + (-0.692 + 0.721i)T \)
37 \( 1 + (-0.987 + 0.160i)T \)
41 \( 1 + (0.568 - 0.822i)T \)
43 \( 1 + (-0.996 - 0.0804i)T \)
47 \( 1 + (0.987 + 0.160i)T \)
53 \( 1 + (-0.948 + 0.316i)T \)
59 \( 1 + (0.845 + 0.534i)T \)
61 \( 1 + (-0.354 + 0.935i)T \)
67 \( 1 + (-0.970 + 0.239i)T \)
71 \( 1 + (0.748 + 0.663i)T \)
73 \( 1 + (0.0402 - 0.999i)T \)
83 \( 1 + (-0.845 + 0.534i)T \)
89 \( 1 + (-0.748 + 0.663i)T \)
97 \( 1 + (0.354 - 0.935i)T \)
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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

−20.09066309395712526555302635393, −19.39488463892284471597481013258, −18.5663961186648963597206644466, −18.01908661403166215485455765906, −17.21431307205127408780277222385, −16.07623253865614290163509984642, −15.36798066418019836148284410438, −14.845814244880799896405881939208, −14.12273524710406403474676816466, −13.1165397393028546276624725141, −12.690321587545885770661239110580, −11.83913993030539206850545017346, −10.94638209716583574809097954325, −9.8235136667088736029677607323, −9.15231375686159718972305178863, −8.60070516823281973120298003797, −7.78647618565830577761908347808, −6.82203973411307262557359037825, −6.14679532501444198512595186331, −5.10094692457422752075311124470, −3.996913559304097006203580748212, −3.20701282169990090737697519460, −2.34922402889701678720530767537, −1.5651561309348798057885651029, −0.26298761988761434398134356667, 1.30892859768236912715272995229, 1.96813685063708891171663834833, 3.29666248018808746304248935386, 3.90927639659332083679761151752, 4.49197556128788600783038613632, 5.73926752014281668986387996220, 7.01513327556540381511058934464, 7.233871064066906222591567612617, 8.37828553739955299624820633050, 9.22255679526253847878100998299, 9.65440179807773089181266530519, 10.592250985795662699505983496524, 11.4153423910542801772472519832, 12.283602144333953056150894645543, 13.40569775832836470394371774192, 13.91194514357653865480850077263, 14.348115765217567186720823010, 15.32713701374727680167553520897, 16.15729765340878261124574811676, 16.677340570762691084213618231303, 17.55092081507912196286297756526, 18.533683107241495417258470102551, 19.3557447245924842538933030908, 19.825454854969394414949972984524, 20.52859820528802511949775639940

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