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Properties

Label 1-4560-4560.1013-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.818 - 0.574i$
Analytic cond. $21.1765$
Root an. cond. $21.1765$
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  + (0.866 − 0.5i)7-s + (0.866 + 0.5i)11-s + (0.766 − 0.642i)13-s + (0.984 + 0.173i)17-s + (0.342 − 0.939i)23-s + (0.984 − 0.173i)29-s + (−0.5 − 0.866i)31-s + 37-s + (0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.984 + 0.173i)47-s + (0.5 − 0.866i)49-s + (−0.939 − 0.342i)53-s + (0.984 + 0.173i)59-s + (−0.342 + 0.939i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)7-s + (0.866 + 0.5i)11-s + (0.766 − 0.642i)13-s + (0.984 + 0.173i)17-s + (0.342 − 0.939i)23-s + (0.984 − 0.173i)29-s + (−0.5 − 0.866i)31-s + 37-s + (0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.984 + 0.173i)47-s + (0.5 − 0.866i)49-s + (−0.939 − 0.342i)53-s + (0.984 + 0.173i)59-s + (−0.342 + 0.939i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.818 - 0.574i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.818 - 0.574i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.452728876 - 0.7753589686i\)
\(L(\frac12)\) \(\approx\) \(2.452728876 - 0.7753589686i\)
\(L(1)\) \(\approx\) \(1.403883423 - 0.1684361771i\)
\(L(1)\) \(\approx\) \(1.403883423 - 0.1684361771i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (0.984 - 0.173i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.984 + 0.173i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.642 - 0.766i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.984 - 0.173i)T \)
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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

−18.244613537918448338455508030624, −17.668389572753609674457313306965, −17.00130636956025356418828514338, −16.18324689945966252315222725463, −15.77429860657240719156819834260, −14.72533474005437696479036133455, −14.28842956327909960454721447084, −13.81951628877685506904581134175, −12.802857742580277810886493036950, −12.12982974244761874655393708322, −11.338366095099161206599347955672, −11.16314997794863754781544231037, −10.07309986560348524453694842828, −9.23205697839899733235120500754, −8.761776424354549222615562638959, −8.0252900986358772930194765620, −7.29424684975401324421554460346, −6.3972548003575759723534982614, −5.76510817974641499398364559449, −5.03846533446775759911584858146, −4.19555104371907303126182001239, −3.46315132626380266799303166162, −2.633135447473490425583610192895, −1.489419493126561375443576919034, −1.13444207913141928929826897270, 0.85267147462809994045018513090, 1.37669410673872032180116982010, 2.41053002785221538931685429025, 3.34045610068619577463648917419, 4.18832867610772442199293894030, 4.667014950528300238225933208895, 5.67521962497404089246880182160, 6.30310153323300224324829548005, 7.166027717318928287001404968646, 7.91542274373077621148216018121, 8.37655673449948854350671876502, 9.309376169167106078836812598280, 10.0028583421748520577339871720, 10.7651244149579018387523197858, 11.29163297421458685128507343215, 12.07236447152976328134149637349, 12.73943980521814354302158844800, 13.46800187359000989259587252955, 14.322258306208291204469523317968, 14.67486798968469930110294770945, 15.329172823054116235505363532140, 16.40449823292204187287845872364, 16.70569736037251873816671352856, 17.70500811828096167711757927374, 17.90529374942807014183621594847

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