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Properties

Label 1-920-920.517-r0-0-0
Degree $1$
Conductor $920$
Sign $0.971 - 0.235i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
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  + (0.281 − 0.959i)3-s + (0.989 + 0.142i)7-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (0.989 − 0.142i)13-s + (0.755 + 0.654i)17-s + (0.654 + 0.755i)19-s + (0.415 − 0.909i)21-s + (−0.755 + 0.654i)27-s + (−0.654 + 0.755i)29-s + (−0.959 + 0.281i)31-s + (0.989 − 0.142i)33-s + (−0.540 + 0.841i)37-s + (0.142 − 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)3-s + (0.989 + 0.142i)7-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (0.989 − 0.142i)13-s + (0.755 + 0.654i)17-s + (0.654 + 0.755i)19-s + (0.415 − 0.909i)21-s + (−0.755 + 0.654i)27-s + (−0.654 + 0.755i)29-s + (−0.959 + 0.281i)31-s + (0.989 − 0.142i)33-s + (−0.540 + 0.841i)37-s + (0.142 − 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.971 - 0.235i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.971 - 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.940947527 - 0.2317542883i\)
\(L(\frac12)\) \(\approx\) \(1.940947527 - 0.2317542883i\)
\(L(1)\) \(\approx\) \(1.352419907 - 0.2195463320i\)
\(L(1)\) \(\approx\) \(1.352419907 - 0.2195463320i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 + (0.989 + 0.142i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (0.989 - 0.142i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + (-0.540 + 0.841i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (-0.281 + 0.959i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.989 + 0.142i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (-0.959 + 0.281i)T \)
67 \( 1 + (0.909 + 0.415i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.755 + 0.654i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (0.540 - 0.841i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.540 - 0.841i)T \)
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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

−21.66673623462868778621612050755, −21.17753582357408422806515229742, −20.47719310244060660041156730184, −19.77153378704744564716200049066, −18.71888789545548280032917716290, −17.98767856149300730884415184020, −16.91977370537633239314729987510, −16.38530824304352447574222755381, −15.54199734797058841813480001440, −14.74134475792216079482137662059, −13.930419850177669782249841311820, −13.507966985992582155568763998266, −11.914007128403034161533318096886, −11.22515255812745737678343027171, −10.74382062285155070122627867381, −9.567007699427452226140897706693, −8.88031904883449872229689344260, −8.142487932867680140655218101145, −7.199356500917524114434610579046, −5.76178289755128308402335314028, −5.24505525994614965341382847690, −4.07753414840195500341217991954, −3.48166567814225049643783550073, −2.30363246176325541774366805282, −0.95339356498517494854228233470, 1.405930294693918563522883752407, 1.66399547453165204232280976329, 3.09493541348335643332595864821, 4.038015389087062678393629007742, 5.32555367025803166338885097384, 6.06547488775831719234160820830, 7.16278451103234993161474524543, 7.83209875241373120746489889834, 8.56358114990049338982478456552, 9.448008417948958303511743532302, 10.64709747414278293421881968910, 11.49273272972739836313287203972, 12.2911980949062816841803811894, 12.89285250413903753803899816188, 14.00031213265549242158808412083, 14.51004436789483365604995278711, 15.219969839473726593885655384579, 16.4421722040210679191766630868, 17.3271045745691623341069669038, 18.05411150887202543804792858372, 18.52915523419769547619770933743, 19.43666536092229578415643400920, 20.46768610739361955657509739431, 20.68626574168011188975746341016, 21.83807523447061698746891959710

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