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Properties

Label 1-1015-1015.739-r1-0-0
Degree $1$
Conductor $1015$
Sign $0.0995 + 0.995i$
Analytic cond. $109.076$
Root an. cond. $109.076$
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  + (−0.294 + 0.955i)2-s + (−0.997 + 0.0747i)3-s + (−0.826 − 0.563i)4-s + (0.222 − 0.974i)6-s + (0.781 − 0.623i)8-s + (0.988 − 0.149i)9-s + (−0.149 + 0.988i)11-s + (0.866 + 0.5i)12-s + (0.623 − 0.781i)13-s + (0.365 + 0.930i)16-s + (0.866 − 0.5i)17-s + (−0.149 + 0.988i)18-s + (−0.997 − 0.0747i)19-s + (−0.900 − 0.433i)22-s + (0.733 + 0.680i)23-s + (−0.733 + 0.680i)24-s + ⋯
L(s)  = 1  + (−0.294 + 0.955i)2-s + (−0.997 + 0.0747i)3-s + (−0.826 − 0.563i)4-s + (0.222 − 0.974i)6-s + (0.781 − 0.623i)8-s + (0.988 − 0.149i)9-s + (−0.149 + 0.988i)11-s + (0.866 + 0.5i)12-s + (0.623 − 0.781i)13-s + (0.365 + 0.930i)16-s + (0.866 − 0.5i)17-s + (−0.149 + 0.988i)18-s + (−0.997 − 0.0747i)19-s + (−0.900 − 0.433i)22-s + (0.733 + 0.680i)23-s + (−0.733 + 0.680i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0995 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0995 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1015\)    =    \(5 \cdot 7 \cdot 29\)
Sign: $0.0995 + 0.995i$
Analytic conductor: \(109.076\)
Root analytic conductor: \(109.076\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1015} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1015,\ (1:\ ),\ 0.0995 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8101888125 + 0.7331519152i\)
\(L(\frac12)\) \(\approx\) \(0.8101888125 + 0.7331519152i\)
\(L(1)\) \(\approx\) \(0.6154919875 + 0.3057703379i\)
\(L(1)\) \(\approx\) \(0.6154919875 + 0.3057703379i\)

Euler product

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

−21.287979463981809432844373537214, −20.74574320734279216858469056171, −19.36117559979764092432446903196, −18.88248118725780891364671849305, −18.37592587745970989578208013011, −17.2923562970265425233594349586, −16.7642796890002810934839551538, −16.14689814764573900274625688967, −14.867774448499692490591679810818, −13.76548851816924448099685809991, −13.10505169763428428836660883037, −12.25569210843277200141111415626, −11.61468543933483946027937970338, −10.72279098651183530135736935351, −10.43834652037964994181273473943, −9.20023850584789939499322340083, −8.47269135460536082831513850450, −7.456873985271122176182645341004, −6.30010305019998053548675397259, −5.54781062103954435782966521800, −4.408709318528742776529442790, −3.75174849462050450313167884199, −2.48735678998184150862634394143, −1.3469956661053013662783298515, −0.53548112988774182850820963184, 0.63919642577694593810978547705, 1.59875537028617823140079230158, 3.40691508513716690290858456487, 4.637298542245635617566758617261, 5.14264838600522143810075377617, 6.08231211242378282416072109669, 6.81879962520619043852587368577, 7.602879675788587108902493764172, 8.49386581473470547427583485747, 9.664052843324958792417760649977, 10.19980341260461795947393225166, 11.026406643206498326678406590393, 12.1462033449578715673336853892, 12.93056531501533184131436073004, 13.65921194882450669835581054925, 14.95738828816944747067205402220, 15.34513934827557484312303000791, 16.18092396459714376088956127523, 17.00844073241153861586789525262, 17.502681147837712991814104242823, 18.293149263041478704963931432147, 18.84420065476897035557274995794, 19.94162355754351841190292736697, 20.98613758443243163909487144112, 21.811498152962513216124897827264

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