Properties

Label 1-1015-1015.787-r0-0-0
Degree $1$
Conductor $1015$
Sign $0.939 - 0.342i$
Analytic cond. $4.71364$
Root an. cond. $4.71364$
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  + (−0.563 − 0.826i)2-s + (−0.149 + 0.988i)3-s + (−0.365 + 0.930i)4-s + (0.900 − 0.433i)6-s + (0.974 − 0.222i)8-s + (−0.955 − 0.294i)9-s + (−0.955 + 0.294i)11-s + (−0.866 − 0.5i)12-s + (−0.974 − 0.222i)13-s + (−0.733 − 0.680i)16-s + (0.866 − 0.5i)17-s + (0.294 + 0.955i)18-s + (0.988 − 0.149i)19-s + (0.781 + 0.623i)22-s + (−0.997 − 0.0747i)23-s + (0.0747 + 0.997i)24-s + ⋯
L(s)  = 1  + (−0.563 − 0.826i)2-s + (−0.149 + 0.988i)3-s + (−0.365 + 0.930i)4-s + (0.900 − 0.433i)6-s + (0.974 − 0.222i)8-s + (−0.955 − 0.294i)9-s + (−0.955 + 0.294i)11-s + (−0.866 − 0.5i)12-s + (−0.974 − 0.222i)13-s + (−0.733 − 0.680i)16-s + (0.866 − 0.5i)17-s + (0.294 + 0.955i)18-s + (0.988 − 0.149i)19-s + (0.781 + 0.623i)22-s + (−0.997 − 0.0747i)23-s + (0.0747 + 0.997i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1015\)    =    \(5 \cdot 7 \cdot 29\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(4.71364\)
Root analytic conductor: \(4.71364\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1015} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1015,\ (0:\ ),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7435685563 - 0.1314721320i\)
\(L(\frac12)\) \(\approx\) \(0.7435685563 - 0.1314721320i\)
\(L(1)\) \(\approx\) \(0.6708099469 - 0.04331820492i\)
\(L(1)\) \(\approx\) \(0.6708099469 - 0.04331820492i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.563 - 0.826i)T \)
3 \( 1 + (-0.149 + 0.988i)T \)
11 \( 1 + (-0.955 + 0.294i)T \)
13 \( 1 + (-0.974 - 0.222i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.988 - 0.149i)T \)
23 \( 1 + (-0.997 - 0.0747i)T \)
31 \( 1 + (0.0747 + 0.997i)T \)
37 \( 1 + (0.294 - 0.955i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (0.680 - 0.733i)T \)
53 \( 1 + (0.997 - 0.0747i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + (-0.680 - 0.733i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.563 - 0.826i)T \)
79 \( 1 + (0.955 + 0.294i)T \)
83 \( 1 + (-0.781 + 0.623i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
97 \( 1 + (-0.781 + 0.623i)T \)
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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

−21.91114648585383846922154019351, −20.6439510427937587581391105431, −19.75669458223459042468364617052, −19.10854609397907828323930346525, −18.38250749481993899712031340651, −17.884114291591323803046288658187, −16.93097424662119320310941695949, −16.43513297754635756880569955352, −15.4309167294024975771333728061, −14.51162499756490867844177503294, −13.882472151372053421804699895, −13.080214964836894033385812618343, −12.15500409337092855202269725999, −11.273478264280688333154345396265, −10.21396028394340512696959997731, −9.53402614764784313096403417459, −8.32338474719906881901770032453, −7.7431706077821039982533638203, −7.22873078978037300306559449876, −6.02143642406369403477823796022, −5.613702141818718642405540667864, −4.54368549303432145978646147576, −2.927421367793219885051805960691, −1.87915287031921687613800935992, −0.7608161990670560671371106320, 0.594403955665037645388920794683, 2.2514321821582333617805171620, 3.00165817705586555186899187154, 3.90655148547723250400102244686, 4.94112347007385461080984730176, 5.5353299488844744962250386127, 7.22625116496191150993174090877, 7.90473079808033428518240261037, 8.91415056904538285046775136019, 9.79664067383732663578546946402, 10.17315687673860622225980835692, 10.9790506623192199557818360421, 11.98311747890195762626488371580, 12.40781188016491960755545740259, 13.67455017451329079325903657462, 14.401157884566068353767173057935, 15.52246813398683842863277319186, 16.211480766783224995262323942793, 16.876706465756517580350234844058, 17.86545655078628802485917903057, 18.262207531366998564197462000653, 19.49990917712514356798304163379, 20.08086407531777275564037595569, 20.79619961027043664048376583894, 21.42179500666427037371502300992

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