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Properties

Label 1-4783-4783.1021-r0-0-0
Degree $1$
Conductor $4783$
Sign $0.328 - 0.944i$
Analytic cond. $22.2121$
Root an. cond. $22.2121$
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  + (−0.0164 − 0.999i)2-s + (0.572 + 0.820i)3-s + (−0.999 + 0.0328i)4-s + (0.559 + 0.829i)5-s + (0.810 − 0.585i)6-s + (−0.178 − 0.983i)7-s + (0.0492 + 0.998i)8-s + (−0.345 + 0.938i)9-s + (0.819 − 0.572i)10-s + (−0.343 − 0.939i)11-s + (−0.598 − 0.800i)12-s + (0.935 − 0.352i)13-s + (−0.980 + 0.194i)14-s + (−0.360 + 0.932i)15-s + (0.997 − 0.0656i)16-s + (0.948 − 0.317i)17-s + ⋯
L(s)  = 1  + (−0.0164 − 0.999i)2-s + (0.572 + 0.820i)3-s + (−0.999 + 0.0328i)4-s + (0.559 + 0.829i)5-s + (0.810 − 0.585i)6-s + (−0.178 − 0.983i)7-s + (0.0492 + 0.998i)8-s + (−0.345 + 0.938i)9-s + (0.819 − 0.572i)10-s + (−0.343 − 0.939i)11-s + (−0.598 − 0.800i)12-s + (0.935 − 0.352i)13-s + (−0.980 + 0.194i)14-s + (−0.360 + 0.932i)15-s + (0.997 − 0.0656i)16-s + (0.948 − 0.317i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4783 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4783 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4783\)
Sign: $0.328 - 0.944i$
Analytic conductor: \(22.2121\)
Root analytic conductor: \(22.2121\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4783} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4783,\ (0:\ ),\ 0.328 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.721961912 - 1.224036700i\)
\(L(\frac12)\) \(\approx\) \(1.721961912 - 1.224036700i\)
\(L(1)\) \(\approx\) \(1.229737186 - 0.3708379490i\)
\(L(1)\) \(\approx\) \(1.229737186 - 0.3708379490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4783 \( 1 \)
good2 \( 1 + (-0.0164 - 0.999i)T \)
3 \( 1 + (0.572 + 0.820i)T \)
5 \( 1 + (0.559 + 0.829i)T \)
7 \( 1 + (-0.178 - 0.983i)T \)
11 \( 1 + (-0.343 - 0.939i)T \)
13 \( 1 + (0.935 - 0.352i)T \)
17 \( 1 + (0.948 - 0.317i)T \)
19 \( 1 + (-0.669 - 0.742i)T \)
23 \( 1 + (0.910 + 0.412i)T \)
29 \( 1 + (-0.0348 - 0.999i)T \)
31 \( 1 + (0.622 + 0.782i)T \)
37 \( 1 + (-0.756 - 0.654i)T \)
41 \( 1 + (0.187 + 0.982i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.563 - 0.826i)T \)
53 \( 1 + (-0.102 - 0.994i)T \)
59 \( 1 + (0.739 - 0.673i)T \)
61 \( 1 + (-0.979 + 0.199i)T \)
67 \( 1 + (-0.472 - 0.881i)T \)
71 \( 1 + (0.993 + 0.115i)T \)
73 \( 1 + (-0.201 + 0.979i)T \)
79 \( 1 + (-0.497 + 0.867i)T \)
83 \( 1 + (-0.581 - 0.813i)T \)
89 \( 1 + (0.642 - 0.765i)T \)
97 \( 1 + (-0.757 - 0.652i)T \)
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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

−18.23753717539905739404385511172, −17.55254686809521838569247345928, −17.0033478907710329407916626256, −16.27975071886610867831934819506, −15.48322416627241838594979817334, −14.97276065984135798734004494725, −14.26722532987740148834613958553, −13.629459876354565379863321916571, −12.95578226136906739879338125244, −12.36892838433882367156179563528, −12.152165974822343364364392363048, −10.58093258359490716329756050397, −9.77331643294349612584304303614, −8.96760909743552916621342181587, −8.76833993258187570492225419435, −8.025529142736099541450523713669, −7.32157764703449935105702875036, −6.44013070934272447660106573319, −5.927178483760476361115957749799, −5.334578979521112410135447657675, −4.43208144186529326202213205557, −3.57507432720502027125554339712, −2.54545391367256838596711800593, −1.67100191354262312296117861150, −0.99001609372964335139598302373, 0.604838918636005186504471375451, 1.589222886242357657651411147934, 2.67002445564309822329291706813, 3.175498101016548401869250744625, 3.62236557252676099506720594593, 4.47476718493474043356549546206, 5.33358314961600762200320239073, 6.01689560259486063643883041597, 7.09673433888958828931744998421, 8.00198095674207025344912235888, 8.57969956516325277699679671304, 9.43614561934210946710163721111, 9.97696373428880238635370114507, 10.62766608893674231899271014601, 10.982072749992141379907697881237, 11.52651814975798932164517877520, 12.95902805523391404671123557873, 13.34144354790166997461507000891, 13.98246973549536241706953714353, 14.31944425567561263524543625106, 15.23214218226767323724683034910, 15.95997591051539300111212775559, 16.84000406785610358981533971967, 17.35161127304307057909460468598, 18.1475540885309218986146294958

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