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Properties

Label 1-4029-4029.185-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.975 + 0.220i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (0.822 − 0.568i)2-s + (0.354 − 0.935i)4-s + (0.0603 − 0.998i)5-s + (0.517 + 0.855i)7-s + (−0.239 − 0.970i)8-s + (−0.517 − 0.855i)10-s + (−0.0603 − 0.998i)11-s + (0.354 − 0.935i)13-s + (0.911 + 0.410i)14-s + (−0.748 − 0.663i)16-s + (−0.992 + 0.120i)19-s + (−0.911 − 0.410i)20-s + (−0.616 − 0.787i)22-s + (0.707 + 0.707i)23-s + (−0.992 − 0.120i)25-s + (−0.239 − 0.970i)26-s + ⋯
L(s)  = 1  + (0.822 − 0.568i)2-s + (0.354 − 0.935i)4-s + (0.0603 − 0.998i)5-s + (0.517 + 0.855i)7-s + (−0.239 − 0.970i)8-s + (−0.517 − 0.855i)10-s + (−0.0603 − 0.998i)11-s + (0.354 − 0.935i)13-s + (0.911 + 0.410i)14-s + (−0.748 − 0.663i)16-s + (−0.992 + 0.120i)19-s + (−0.911 − 0.410i)20-s + (−0.616 − 0.787i)22-s + (0.707 + 0.707i)23-s + (−0.992 − 0.120i)25-s + (−0.239 − 0.970i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.975 + 0.220i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.975 + 0.220i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2488862848 - 2.232470398i\)
\(L(\frac12)\) \(\approx\) \(-0.2488862848 - 2.232470398i\)
\(L(1)\) \(\approx\) \(1.187046729 - 1.055643409i\)
\(L(1)\) \(\approx\) \(1.187046729 - 1.055643409i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.822 - 0.568i)T \)
5 \( 1 + (0.0603 - 0.998i)T \)
7 \( 1 + (0.517 + 0.855i)T \)
11 \( 1 + (-0.0603 - 0.998i)T \)
13 \( 1 + (0.354 - 0.935i)T \)
19 \( 1 + (-0.992 + 0.120i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.297 - 0.954i)T \)
31 \( 1 + (-0.983 - 0.180i)T \)
37 \( 1 + (0.787 + 0.616i)T \)
41 \( 1 + (-0.998 - 0.0603i)T \)
43 \( 1 + (-0.663 - 0.748i)T \)
47 \( 1 + (-0.120 + 0.992i)T \)
53 \( 1 + (-0.239 - 0.970i)T \)
59 \( 1 + (-0.935 + 0.354i)T \)
61 \( 1 + (0.616 - 0.787i)T \)
67 \( 1 + (0.568 - 0.822i)T \)
71 \( 1 + (0.517 - 0.855i)T \)
73 \( 1 + (-0.911 - 0.410i)T \)
83 \( 1 + (0.935 + 0.354i)T \)
89 \( 1 + (-0.970 - 0.239i)T \)
97 \( 1 + (0.787 + 0.616i)T \)
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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

−18.57668114693227674793965517920, −18.107079555380094229883101797601, −17.28967279332673251392503470789, −16.80180455067686871922870135293, −16.07262834325273380766276285878, −15.16149547050431480442656743320, −14.58466491565144533441497549686, −14.37430512058810552247702228686, −13.445761924495590359710495746581, −12.90388896052268504579305643192, −12.05148112110890558846370493961, −11.21691460213438840826149843867, −10.81648475873803540018190421317, −10.0198597824117309561231440901, −8.95127952392830928976183933098, −8.18846789186832359838892476201, −7.26302144783971310098437776370, −6.92541025138576335016579632304, −6.40422041758526817729633239625, −5.33936835097892619566045822181, −4.50041547125592018802322960702, −4.05760270763857304283925195681, −3.198541084517922696270749221286, −2.28070472038494570893692505210, −1.589476086206287694852887340272, 0.41955119103475543380092086270, 1.3957947932843180460588692605, 2.09404958894832982222968925889, 3.048054174319831425894539517218, 3.73376575388908872163518116280, 4.7045876122846418096565576928, 5.26540196978131408763435612748, 5.84749675574161931602006490253, 6.447531145420038454954208369914, 7.8387188232894580283580197959, 8.39745091986152297192908796481, 9.1189641566949800600738284273, 9.83224459499245907432331740996, 10.828044500648506756784301967800, 11.29915374251757402353181157414, 12.02771431997373497355746026356, 12.66683558533429327813616475745, 13.27670047705348097871205293231, 13.73967126720068379609675257882, 14.775156420278373837228823413075, 15.28340021131471658620997984174, 15.85144739360077784277460406668, 16.673878332111627305410234736319, 17.40044143489705741473293094556, 18.28746715152956500437580256857

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