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Properties

Label 1-783-783.122-r1-0-0
Degree $1$
Conductor $783$
Sign $0.181 + 0.983i$
Analytic cond. $84.1450$
Root an. cond. $84.1450$
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  + (0.542 + 0.840i)2-s + (−0.411 + 0.911i)4-s + (−0.921 + 0.388i)5-s + (−0.411 − 0.911i)7-s + (−0.988 + 0.149i)8-s + (−0.826 − 0.563i)10-s + (−0.661 − 0.749i)11-s + (−0.853 + 0.521i)13-s + (0.542 − 0.840i)14-s + (−0.661 − 0.749i)16-s + (−0.5 − 0.866i)17-s + (−0.0747 + 0.997i)19-s + (0.0249 − 0.999i)20-s + (0.270 − 0.962i)22-s + (−0.456 − 0.889i)23-s + ⋯
L(s)  = 1  + (0.542 + 0.840i)2-s + (−0.411 + 0.911i)4-s + (−0.921 + 0.388i)5-s + (−0.411 − 0.911i)7-s + (−0.988 + 0.149i)8-s + (−0.826 − 0.563i)10-s + (−0.661 − 0.749i)11-s + (−0.853 + 0.521i)13-s + (0.542 − 0.840i)14-s + (−0.661 − 0.749i)16-s + (−0.5 − 0.866i)17-s + (−0.0747 + 0.997i)19-s + (0.0249 − 0.999i)20-s + (0.270 − 0.962i)22-s + (−0.456 − 0.889i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(783\)    =    \(3^{3} \cdot 29\)
Sign: $0.181 + 0.983i$
Analytic conductor: \(84.1450\)
Root analytic conductor: \(84.1450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{783} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 783,\ (1:\ ),\ 0.181 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7151876429 + 0.5952857112i\)
\(L(\frac12)\) \(\approx\) \(0.7151876429 + 0.5952857112i\)
\(L(1)\) \(\approx\) \(0.7494234797 + 0.3588767557i\)
\(L(1)\) \(\approx\) \(0.7494234797 + 0.3588767557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.542 + 0.840i)T \)
5 \( 1 + (-0.921 + 0.388i)T \)
7 \( 1 + (-0.411 - 0.911i)T \)
11 \( 1 + (-0.661 - 0.749i)T \)
13 \( 1 + (-0.853 + 0.521i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.0747 + 0.997i)T \)
23 \( 1 + (-0.456 - 0.889i)T \)
31 \( 1 + (0.797 + 0.603i)T \)
37 \( 1 + (-0.365 + 0.930i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (-0.921 - 0.388i)T \)
47 \( 1 + (0.980 - 0.198i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.411 + 0.911i)T \)
67 \( 1 + (-0.318 - 0.947i)T \)
71 \( 1 + (0.988 + 0.149i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (0.853 + 0.521i)T \)
83 \( 1 + (-0.698 + 0.715i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (-0.270 - 0.962i)T \)
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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

−21.95318707077662656441387937882, −21.22025326473032482054904147540, −20.19270529361193098714862390320, −19.66970370250354496049499758395, −19.08313191090786900390928487777, −18.09681320629403858880604812413, −17.316327710725444759775732902645, −15.861989311034500358389813528780, −15.2801019298295613644790178812, −14.88372867378353675663043603305, −13.38705285630749344993795414546, −12.80279076128677587637746083738, −12.127169266680637652358207143337, −11.483840014999299884846730437959, −10.43936323296943588259409360302, −9.60392740536643738388311214296, −8.7411904864901785341027713558, −7.76717334310325216675268257860, −6.58800387997876590531369699258, −5.39118579185973626985084377272, −4.79165624733443217445091444437, −3.77997114771261963351874018449, −2.772614480015416205267077059095, −1.95042812913803974461720836923, −0.38007199803152598340889088870, 0.44750576134296578767727819778, 2.62942445472623744801156598024, 3.49731324843011164402560527469, 4.32430412532589249865582195047, 5.1518315650019748317535612533, 6.50236696031657635659855415663, 7.00171232801205125769538723945, 7.89533609053395409209883376994, 8.55259031116008222653085226082, 9.87664491552766446461445050703, 10.78105545232482998652750506572, 11.87668449385751198411198913220, 12.45566307048739641350410606757, 13.71119595593546476666682469664, 14.01593803326161167183684974654, 15.07630068932144909855246709085, 15.84069513024825889050542546118, 16.50602398343166953091840542911, 17.0899184932158171866281655526, 18.37346703368456763697871678011, 18.89415668442841888149661468676, 20.011790782773017541824555016409, 20.71724737943118665735291050424, 21.80814430392192491642195112226, 22.53523325738799195209310698375

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